Classroom Tips and Techniques: Trigonometric Parametrization of an Ellipse Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= A circle centered at the origin is easily parametrized trigonometrically as 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, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the radius of the circle. If the center of the circle is translated to the point LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtaUdGJDYlUSJoRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJrRidGNEY3Rj5GPkY+, the trigonometric parametrization readily extends to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY2LUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSI9RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZULUYsNiVRImhGJ0YvRjItRkA2LVEiK0YnRj1GQ0ZGRkhGSkZMRk5GUC9GU1EsMC4yMjIyMjIyZW1GJy9GVkZobi1GLDYlUSJhRidGL0YyLUZANi1RIn5GJ0Y9RkNGRkZIRkpGTEZORlAvRlNRJjAuMGVtRicvRlZGYW8tRiw2JVEkY29zRicvRjBGRUY9RjUtRkA2LVEiLEYnRj1GQy9GR0YxRkhGSkZMRk5GUEZgby9GVlEsMC4zMzMzMzMzZW1GJy1GLDYlUSJ5RidGL0YyRjVGPy1GLDYlUSJrRidGL0YyRlpGam5GXW8tRiw2JVEkc2luRidGZm9GPUY1Rj0=. An ellipse whose standard form in Cartesian coordinates is 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 can also be parametrized trigonometrically as 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 since 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 In this month's article, we discuss a trigonometric parametrization for the ellipse whose Cartesian equation contains an LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjeHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn-term, indicating that the axes of the ellipse are rotated with respect to the coordinate axes.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRLkxpbmVhckFsZ2VicmFGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1JI21vR0YkNi1RIjpGJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRj8vJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVkY9RkA= QyQtSSV3aXRoRzYiNiNJJnBsb3RzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlISIi LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= One does not get far into a Calculus course with encountering the trigonometric parametrization of a circle. Thus, the circle 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= is readily parametrized as 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. Of course, if 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, then the circle is given simply by 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. In either event, the trigonometric parametrization is free of the branch issues that makes the explicit representation 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 so much more awkward to work with. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The trigonometric parametrization of an ellipse with axes parallel to the coordinate axes is nearly as familiar as the trigonometric parametrization of a circle. Thus, the ellipse 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= is readily parametrized as 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. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= To obtain an equivalent trigonometric parametrization of an ellipse whose axes are not parallel to the coordinate axes requires application of both a translation and a rotation, calculations that can be found in a college-level analytic geometry text. For example, I still use the text Analytic Geometry, C. O. Oakley, Barnes & Noble, 1959, that I acquired years ago for a cost of $1.50. Unfortunately, I don't think this material appeared in the syllabi of any of the college math courses I've taught over the years. I did, however, need to make use of the material when exploring (in the pre-computer days) how penalty functions work in constrained optimization problems.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= JSFH

JSFH Consider, then, the quadratic equation 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 whose graph is seen in Figure 1. This figure is obtained as per Table 1 by first loading the Student Precalculus package, then applying the Context Menu to the quadratic equation, selecting Tutors\342\211\273Conic Sections. JSFH Tools\342\211\273Load Package: Student Precalculus with(Student:-Calculus1): Loading Student:-Precalculus with(Student:-Precalculus): 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Figure 1 Conic Sections Tutor applied to the quadratic equation 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 Table 1 Application of the Conic Sections Tutor to 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. The option "Constrained Scaling" has been selected in the Plot Options dialog. The standard form displayed is in the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZwcmltZTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRictRiw2JVEieUYnRi9GMkY1Rjk=-plane whose coordinate axes are aligned with the axes of the ellipse. JSFH Figure 2 shows all the data provided by the Conic Sections Tutor. JSFH class: ellipse eccentricity: .884 semimajor axis (a): 11.5 semiminor axis (b): 5.37 latus rectum: 5.03 angle: .448 ----------------------------- In the xy-plane: vertices: [(.573e-1,-2.90), (20.7,7.03)] foci: [(1.26,-2.32), (19.5,6.46)] center (h,k): (10.4,2.07) directrix: y = -2.08*x+53.6 ----------------------------- In the x'y'-plane: vertices: [(-1.20,-2.64), (21.7,-2.64)] foci: [(.131,-2.64), (20.4,-2.64)] center (h',k'): (10.3,-2.64) directrix: x' = 23.2 Figure 2 Data provided by the Conic Sections Tutor JSFH Maple states that the quadratic defines an ellipse, and gives its parameters in the original LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjeHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn-plane, and in the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZwcmltZTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRictRiw2JVEieUYnRi9GMkY1Rjk=-plane where the coordinate axes are aligned with the axes of the ellipse. In addition to the eccentricity, the foci and vertices are given in both planes, the equation of the directrix is given in the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZwcmltZTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRictRiw2JVEieUYnRi9GMkY1Rjk=-plane, and the angular offset between the two sets of axes is provided. Unfortunately, the Tutor does not have enough room to display the exact form of any of these quantities, which can be large and cumbersome. Hence, the data is reported in floating-point form with an abbreviated number of digits. JSFH Table 2 shows an alternative analysis of the ellipse, this one provided by the geometry package.JSFH JSFH The geometry package has to be loaded with the syntax shown to the right. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRKWdlb21ldHJ5RidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkAtSSNtb0dGJDYtUSI6RidGQC8lJmZlbmNlR0Y/LyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlZGPUZA JSFH print(); # input placeholder The ellipse is made known to the package with the syntax shown to the right. The ellipse is named "E" by this command. 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= print(); # input placeholder Below, the detail command provides the results (in exact arithmetic) of Maple's analysis of the ellipse. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEpc2ltcGxpZnlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Ji1GLDYlUSdkZXRhaWxGJ0YvRjItRjY2JC1GIzYlLUYsNiVRIkVGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGR0ZERkdGR0ZERkc=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LUknbXRhYmxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNjstSSRtdHJHRiQ2Jy1JJG10ZEdGJDYoLUkmbXRleHRHRiQ2JFEzbmFtZX5vZn50aGV+b2JqZWN0RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUpcm93YWxpZ25HUSFGJy8lLGNvbHVtbmFsaWduR0Y6LyUrZ3JvdXBhbGlnbkdGOi8lKHJvd3NwYW5HUSIxRicvJStjb2x1bW5zcGFuR0ZBLUYvNigtSSNtaUdGJDYlUSJFRicvJSdpdGFsaWNHUSV0cnVlRicvRjZRJ2l0YWxpY0YnRjhGO0Y9Rj9GQkY4RjtGPS1GLDYnLUYvNigtRjI2JFEzZm9ybX5vZn50aGV+b2JqZWN0RidGNUY4RjtGPUY/RkItRi82KC1GRzYlUSplbGxpcHNlMmRGJ0ZKRk1GOEY7Rj1GP0ZCRjhGO0Y9LUYsNictRi82KC1GMjYkUSdjZW50ZXJGJ0Y1RjhGO0Y9Rj9GQi1GLzYoLUkobWZlbmNlZEdGJDYmLUklbXJvd0dGJDYoLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRJDYxM0YnRjUtRmhvNiRRIzU5RidGNS8lLmxpbmV0aGlja25lc3NHRkEvJStkZW5vbWFsaWduR0Zbby8lKW51bWFsaWduR0Zbby8lKWJldmVsbGVkR1EmZmFsc2VGJy1JI21vR0YkNi1RIixGJ0Y1LyUmZmVuY2VHRmZwLyUqc2VwYXJhdG9yR0ZMLyUpc3RyZXRjaHlHRmZwLyUqc3ltbWV0cmljR0ZmcC8lKGxhcmdlb3BHRmZwLyUubW92YWJsZWxpbWl0c0dGZnAvJSdhY2NlbnRHRmZwLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRmVvNigtRmhvNiRRJDEyMkYnRjVGW3BGXnBGYHBGYnBGZHAvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUpcmVhZG9ubHlHRmZwRjVGNS8lJW9wZW5HUScmbHNxYjtGJy8lJmNsb3NlR1EnJnJzcWI7RidGOEY7Rj1GP0ZCRjhGO0Y9LUYsNictRi82KC1GMjYkUSVmb2NpRidGNUY4RjtGPUY/RkItRi82KC1GX282Ji1GYm82KC1GX282Ji1GYm82KC1GYm82KEZkby1GaHA2LVEoJm1pbnVzO0YnRjVGW3EvRl5xRmZwRl9xRmFxRmNxRmVxRmdxL0ZqcVEsMC4yMjIyMjIyZW1GJy9GXXJGZ3QtRmVvNigtRmJvNiotRmhvNiRRIjRGJ0Y1LUZocDYtUTEmSW52aXNpYmxlVGltZXM7RidGNUZbcUZldEZfcUZhcUZjcUZlcUZncUZpcS9GXXJGW3ItSSZtc3FydEdGJDYjLUZobzYkUSYxMzk0NUYnRjVGYHUtSSVtc3VwR0YkNiUtRmhvNiRRIzQxRidGNS1GZW82KC1GaG82JEZBRjVGXXVGXnBGYHBGYnAvRmVwRkwvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRmB1LUZldTYjLUZibzYmLUZobzYkUSUzMzYyRidGNUZidC1GYm82Ji1GaG82JFEkMzI4RidGNUZgdS1GZXU2I0ZddkY1RjVGNS1GaG82JFEmMTIwOTVGJ0Y1Rl5wRmBwRmJwRmRwRmJ0LUZlbzYoLUZibzYoRmR1RmB1LUZbdjYlRl12LUZlbzYoLUZobzYkUSIzRidGNUZddUZecEZgcEZicEZkdkZldkZgdUZodkY1RmZ3Rl5wRmBwRmJwRmRwRjVGZ3AtRmJvNictRmhwNi1RKiZ1bWludXMwO0YnRjVGW3FGZXRGX3FGYXFGY3FGZXFGZ3FGZnRGaHQtRmVvNigtRmJvNihGZHVGYHVGanVGYHVGaHZGNS1GaG82JFElMjQxOUYnRjVGXnBGYHBGYnBGZHAtRmhwNi1RJyZwbHVzO0YnRjVGW3FGZXRGX3FGYXFGY3FGZXFGZ3FGZnRGaHRGX3JGNUZkckZnckY1RjVGaXJGXHNGZ3AtRl9vNiYtRmJvNigtRmJvNihGZG9GYHlGaXRGYHlGaXdGNUZncC1GYm82JkZpeEZgeUZfckY1RmRyRmdyRjVGNUZpckZcc0ZkckZnckY1RjVGaXJGXHNGOEY7Rj1GP0ZCRjhGO0Y9LUYsNictRi82KC1GMjYkUTlsZW5ndGh+b2Z+dGhlfm1ham9yfmF4aXNGJ0Y1RjhGO0Y9Rj9GQi1GLzYoLUZibzYlRmZ4LUZlbzYoLUZibzYoLUZobzYkUSIyRidGNUZgdUZkdUZgdS1GZXU2I0ZqekY1LUZibzYmLUZfbzYkLUZibzYnRmZ4LUZobzYkUSMxMEYnRjVGYHlGZHdGNUY1RmB1LUZldTYjLUZibzYmRmVbbEZgeUZkd0Y1RjVGXnBGYHBGYnBGZHBGNUY4RjtGPUY/RkJGOEY7Rj0tRiw2Jy1GLzYoLUYyNiRROWxlbmd0aH5vZn50aGV+bWlub3J+YXhpc0YnRjVGOEY7Rj1GP0ZCLUYvNigtRmVvNihGaHotRmJvNiYtRl9vNiRGaltsRjVGYHUtRmV1NiMtRmJvNiZGZVtsRmJ0RmR3RjVGNUZecEZgcEZicEZkcEY4RjtGPUY/RkJGOEY7Rj0tRiw2Jy1GLzYoLUYyNiRROGVxdWF0aW9ufm9mfnRoZX5lbGxpcHNlRidGNUY4RjtGPUY/RkItRi82KC1GYm82Ji1GYm82Li1GYm82JkZheEZgdS1GW3Y2JS1GRzYlUSJ4RidGSkZNRmp6RmV2RjVGYnQtRmJvNiYtRmhvNiRRIjVGJ0Y1RmB1LUZibzYmRmBebEZgdS1GRzYlUSJ5RidGSkZNRjVGNUZgeS1GYm82Ji1GaG82JFEiN0YnRjVGYHUtRlt2NiVGal5sRmp6RmV2RjVGYnQtRmJvNiYtRmhvNiRRIzUyRidGNUZgdUZgXmxGNUZgeS1GYm82Ji1GaG82JFEjMjNGJ0Y1RmB1RmpebEY1RmB5RmVbbEY1LUZocDYtUSkmZXF1YWxzO0YnRjVGW3FGZXRGX3FGYXFGY3FGZXFGZ3EvRmpxUSwwLjI3Nzc3NzhlbUYnL0ZdckZiYGwtRmhvNiRGZ3ZGNUY1RjhGO0Y9Rj9GQkY4RjtGPS8lJmFsaWduR1ElYXhpc0YnL0Y5USliYXNlbGluZUYnL0Y8USVsZWZ0RicvRj5RJ3xmcmxlZnR8aHJGJy8lL2FsaWdubWVudHNjb3BlR0ZMLyUsY29sdW1ud2lkdGhHUSVhdXRvRicvJSZ3aWR0aEdGY2FsLyUrcm93c3BhY2luZ0dRJjEuMGV4RicvJS5jb2x1bW5zcGFjaW5nR1EmMC44ZW1GJy8lKXJvd2xpbmVzR1Elbm9uZUYnLyUsY29sdW1ubGluZXNHRl5ibC8lJmZyYW1lR0ZeYmwvJS1mcmFtZXNwYWNpbmdHUSwwLjRlbX4wLjVleEYnLyUqZXF1YWxyb3dzR0ZmcC8lLWVxdWFsY29sdW1uc0dGZnAvJS1kaXNwbGF5c3R5bGVHRmZwLyUlc2lkZUdRJnJpZ2h0RicvJTBtaW5sYWJlbHNwYWNpbmdHRltibA==

The angle of rotation between the sets of axes is not given, and all the data is given just for the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjeHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn-plane. That this is so can be verified by extracting the coordinates of the foci, writing them in floating-point form, and comparing the results with those listed in Figure 2. See the calculation below. 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

NyQ3JCQiKy8uPWc3ISIqJCErRCw5P0JGJjckJCIrciFbPiY+ISIpJCIrW0x0YmtGJg==

Table 2 Analysis of a quadratic via the ellipse command in the geometry package JSFH

Student:-Precalculus:-ConicsTutor( ); Student:-Precalculus:-ConicsTutor( );

JSFH The general quadratic equation in two variables (say, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=) can be written in the form 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 JSFH Compute LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEnJiM5NjM7RidGL0YyRjI=, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEoJnNpZ21hO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSI9RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkktRiw2JVEiYUYnL0YwUSV0cnVlRicvRjNRJ2l0YWxpY0YnLUY2Ni1RIitGJ0YyRjlGO0Y9Rj9GQUZDRkUvRkhRLDAuMjIyMjIyMmVtRicvRktGVy1GLDYlUSJjRidGT0ZRRjI=, 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 The quantities LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjM7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIi9GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdRJXRydWVGJy8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4xNjY2NjY3ZW1GJy8lJ3JzcGFjZUdGSi1JI21uR0YkNiRRIjhGJ0YyRjI= are invariant under rotations. If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NjY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIjxGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjBGJ0YyRjI=, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RKyZOb3RFcXVhbDtGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjBGJ0YyRjI=, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjM7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy are of opposite signs, the quadratic equation defines an ellipse. The center of the ellipse, namely, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtaUdGJDYlUSJoRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJrRidGNEY3Rj5GPi1GMTYjUSFGJ0Y+, is given by 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 and 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 JSFH Translating the conic to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjdXZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn-coordinates whose origin is at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtaUdGJDYlUSJoRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJrRidGNEY3Rj5GPkY+ puts the ellipse into the form LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY0LUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1JJW1zdXBHRiQ2JS1GLDYlUSJ1RidGL0YyLUYjNiUtSSNtbkdGJDYkUSIyRidGOUYvRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUY2Ni1RIitGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GXG8tRiw2JVEiYkYnRi9GMkY1RlJGNS1GLDYlUSJ2RidGL0YyRmhuLUYsNiVRImNGJ0YvRjJGNS1GUDYlRmFvRlVGZW4tRjY2LVEoJm1pbnVzO0YnRjlGO0Y+RkBGQkZERkZGSEZbb0Zdby1GLDYlUScmIzk2MTtGJy9GMEY9RjktRjY2LVEiPUYnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yNzc3Nzc4ZW1GJy9GTkZkcC1GWDYkRmduRjlGOQ== JSFH where 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. Rotation of the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjdXZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn-axes through an angle LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy is defined by 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 JSFHJSFH removes the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjdXZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn-term. (If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJjRidGL0YyRjk=, then 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 so 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.) We let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy be given by 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 where the two-argument arctangent function is used to place LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy in the proper quadrant. Then, we define LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEkc2luRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRJyYjOTUyO0YnRi9GMkYyRjItRiw2I1EhRidGPS1JI21vR0YkNi1RKCZlcXVpdjtGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVC1GLDYlUSJTRicvRjBRJXRydWVGJy9GM1EnaXRhbGljRidGMg== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEkY29zRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRJyYjOTUyO0YnRi9GMkYyRjItRiw2I1EhRidGPS1JI21vR0YkNi1RKCZlcXVpdjtGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVC1GLDYlUSJDRicvRjBRJXRydWVGJy9GM1EnaXRhbGljRidGMg== JSFH The LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjdXZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn-coordinate axes are rotated to the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjWFlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn-axes, parallel to the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjeHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn-coordinate axes, by the transformation 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 JSFH The result is the equation 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, where 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 and 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 The standard form 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 is obtained by defining LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnJiM5NDU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1GIzYlLUYsNiNRIUYnLUkmbXNxcnRHRiQ2Iy1GIzYkLUkmbWZyYWNHRiQ2KC1GLDYmUScmIzk2MTtGJ0YvLyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR1EldHJ1ZUYnRjItRiM2Ji1GLDYmUSJQRicvRjBGaG5GZm4vRjNRJ2l0YWxpY0YnRl5vRmZuRl9vLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zmby8lKWJldmVsbGVkR0YxRjJGMkZORjI= and 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 Consequently, the given ellipse is parametrized by the equations LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tRiw2I1EhRidGPQ== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEiPUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1GLDYtUSIrRidGL0YyRjVGN0Y5RjtGPUY/L0ZCUSwwLjIyMjIyMjJlbUYnL0ZFRlMtRkc2JVEiaEYnRkpGTUYv LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEiPUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1GLDYtUSIrRidGL0YyRjVGN0Y5RjtGPUY/L0ZCUSwwLjIyMjIyMjJlbUYnL0ZFRlMtRkc2JVEia0YnRkpGTUYv LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 JSFH For convenience, we summarize the requisite calculations in Table 3.JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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Calculations leading to a trigonometric parametrization of an ellipse JSFH

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LC4qJEkieEc2IiIiIyIiJComRiQiIiJJInlHRiVGKSEiJiokRipGJiIiKEYkISNfRioiI0IiIzVGKQ==

Assign the coefficients. The command used provides the Taylor coefficient of the requisite power of the variables. Since the command is applied to a quadratic, the second degree Taylor polynomial is the quadratic, so the coefficients of the quadratic are precisely the coefficients of the Taylor polynomials of the appropriate degrees. Qy4+SSJhRzYiLUkpY29lZnRheWxHRiU2JUkicUdGJS83JEkieEdGJUkieUdGJTckIiIhRi83JCIiI0YvIiIiPkkiYkdGJS1GJzYlRilGKjckRjJGMkYyPkkiY0dGJS1GJzYlRilGKjckRi9GMUYyPkkiZEdGJS1GJzYlRilGKjckRjJGL0YyPkkiZUdGJS1GJzYlRilGKjckRi9GMkYyPkkiZkdGJS1GJzYlRilGKkYuRjI=

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Compute LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy. PkkkcGhpRzYiLCYqJEkiYkdGJCIiIyIiIiomSSJhR0YkRilJImNHRiRGKSEiJQ==

ISNm

Compute LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy. PkkkcmhvRzYiLCQqJkkmZGVsdGFHRiQiIiJJJHBoaUdGJCEiIiNGKCIiIw==

IyImWFIiIiNm

Compute LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. PkkiaEc2IiomLCYqJkkiY0dGJCIiIkkiZEdGJEYpIiIjKiZJImJHRiRGKUkiZUdGJEYpISIiRilJJHBoaUdGJEYv

IyIkOCciI2Y=

Compute LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Pkkia0c2IiomLCYqJkkiYUdGJCIiIkkiZUdGJEYpIiIjKiZJImJHRiRGKUkiZEdGJEYpISIiRilJJHBoaUdGJEYv

IyIkQSIiI2Y=

Since LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjNGJ0Y5LUY2Ni1RKyZOb3RFcXVhbDtGJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUZQNiRRIjdGJ0Y5RjUtRiw2JVEiY0YnRi9GMkY5, compute 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. PkkmdGhldGFHNiIsJC1JJ2FyY3Rhbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYkSSJiR0YkLCZJImFHRiQiIiJJImNHRiQhIiIjRi8iIiM=

LCYtSSdhcmN0YW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyMiIiYiIiUjIiIiIiIjSSNQaUdGJiMhIiJGLw==

Compute LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSRzaW5GJy9GMEY9RjktSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUScmIzk1MjtGJ0ZSRjlGOUY5Rjk=. PkkiU0c2Ii1JJmV2YWxjRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMtSShjb252ZXJ0R0YoNiQtSSRzaW5HRiQ2I0kmdGhldGFHRiRJJGV4cEdGJw==

LCQqJCwmIiVpTCIiIiokIiNUI0YmIiIjIiRHJEYpIyEiIiIjIyk=

Compute LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSRjb3NGJy9GMEY9RjktSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUScmIzk1MjtGJ0ZSRjlGOUY5Rjk=. PkkiQ0c2Ii1JJmV2YWxjRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMtSShjb252ZXJ0R0YoNiQtSSRjb3NHRic2I0kmdGhldGFHRiRJJGV4cEdGJw==

LCQqJCwmIiVpTCIiIiokIiNUI0YmIiIjISRHJEYpI0YmIiMjKQ==

Compute LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. PkkiUEc2Ii1JKnJhZG5vcm1hbEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjLCgqJkkiYUdGJCIiIkkiQ0dGJCIiI0YuKiZJImNHRiRGLkkiU0dGJEYwRi4qKEkiYkdGJEYuRi9GLkYzRi5GLg==

LCYiIiYiIiIqJCIjVCNGJCIiI0Yn

Compute LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. PkkiUUc2Ii1JKnJhZG5vcm1hbEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjLCgqJkkiYUdGJCIiIkkiU0dGJCIiI0YuKiZJImNHRiRGLkkiQ0dGJEYwRi4qKEkiYkdGJEYuRjNGLkYvRi4hIiI=

LCYiIiYiIiIqJCIjVCNGJCIiIyMhIiJGKA==

Compute LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy. PkkmYWxwaGFHNiItSSVzcXJ0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMqJkkkcmhvR0YkIiIiSSJQR0YkISIi LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LCQqJiInYkYjKSMiIiIiIiMsJiIiJkYmKiQiI1RGJUYlIyEiIkYnI0YmIiNm

Compute LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy. PkklYmV0YUc2Ii1JJXNxcnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyomSSRyaG9HRiQiIiJJIlFHRiQhIiI=

LCQqJiInYkYjKSMiIiIiIiMsJiIiJkYmKiQiI1RGJSMhIiJGJ0YsI0YmIiNm

Obtain LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==. PkkjeHRHNiIsKCooSSZhbHBoYUdGJCIiIkkiQ0dGJEYoLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNJInRHRiRGKEYoKihJJWJldGFHRiRGKEkiU0dGJEYoLUkkc2luR0YsRi9GKCEiIkkiaEdGJEYo

LCgqKiInYkYjKSMiIiIiIiMsJiIiJkYmKiQiI1RGJUYlIyEiIkYnLCYiJWlMRiZGKiEkRyRGJS1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSJ0R0Y2RiYjRiYiJVFbKipGJEYlLCZGKUYmRipGLEYsLCZGL0YmRioiJEckRiUtSSRzaW5HRjNGN0YmRjkjIiQ4JyIjZkYm

Obtain LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==. PkkjeXRHNiIsKCooSSZhbHBoYUdGJCIiIkkiU0dGJEYoLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNJInRHRiRGKEYoKihJJWJldGFHRiRGKEkiQ0dGJEYoLUkkc2luR0YsRi9GKEYoSSJrR0YkRig=

LCgqKiInYkYjKSMiIiIiIiMsJiIiJkYmKiQiI1RGJUYlIyEiIkYnLCYiJWlMRiZGKiIkRyRGJS1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSJ0R0Y2RiYjRi0iJVFbKipGJEYlLCZGKUYmRipGLEYsLCZGL0YmRiohJEckRiUtSSRzaW5HRjNGN0YmI0YmRjojIiRBIiIjZkYm

Table 4 Trigonometric parametrization of the quadratic LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY6LUkjbW5HRiQ2JFEiM0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkctSSVtc3VwR0YkNiUtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUYjNiYtRiw2JFEiMkYnRi8vJStleGVjdXRhYmxlR0ZTLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJ0YvLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GMzYtUSgmbWludXM7RidGL0Y2RjlGO0Y9Rj9GQUZDL0ZGUSwwLjIyMjIyMjJlbUYnL0ZJRmFvLUYsNiRRIjVGJ0YvRjJGTUYyLUZONiVRInlGJ0ZRRlQtRjM2LVEiK0YnRi9GNkY5RjtGPUY/RkFGQ0Zgb0Ziby1GLDYkUSI3RidGL0YyLUZLNiVGZm9GVkZqbkZdby1GLDYkUSM1MkYnRi9GMkZNRmlvLUYsNiRRIzIzRidGL0YyRmZvRmlvLUYsNiRRIzEwRidGL0Yv LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= To verify that the parametrization obtain in Table 4 actually represents the original ellipse, we provide Figure 3 in which an implicit plot of the ellipse is superimposed on the parametric plot determined by the parametrization in Table 4. Qyc+JkkicEc2IjYjIiIiLUktaW1wbGljaXRwbG90R0YmNidJInFHRiYvSSJ4R0YmOyEiIiIjQC9JInlHRiY7ISImIiM1L0kmY29sb3JHRiZJJHJlZEdGJi9JKnRoaWNrbmVzc0dGJiIiJkYwPiZGJTYjIiIjLUklcGxvdEdGJjYlNyVJI3h0R0YmSSN5dEdGJi9JInRHRiY7IiIhLCRJI1BpRyUqcHJvdGVjdGVkR0ZAL0Y4SSZibGFja0dGJi9GOyIiJEYwLUkoZGlzcGxheUdGJjYjNyRGJEY+

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

%;

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Figure 3 Implicit plot of the ellipse determined by the given quadratic superimposed on the graph of its parametrization. The implicit plot is in red; the parametric curve, in black. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Although Figure 3 suggests that the recipe in Table 3 is correct, it is useful to derive these results. To this end, we start afresh, with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

LC4qJkkiYUc2IiIiIilJInhHRiUiIiNGJkYmKihJImJHRiVGJkYoRiZJInlHRiVGJkYmKiZJImNHRiVGJilGLEYpRiZGJiomSSJkR0YlRiZGKEYmRiYqJkkiZUdGJUYmRixGJkYmSSJmR0YlRiY=

The first step is the translation 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, implemented as 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

LC4qJkkiYUc2IiIiIiksJkkidUdGJUYmSSJoR0YlRiYiIiNGJkYmKihJImJHRiVGJkYoRiYsJkkidkdGJUYmSSJrR0YlRiZGJkYmKiZJImNHRiVGJilGLkYrRiZGJiomSSJkR0YlRiZGKEYmRiYqJkkiZUdGJUYmRi5GJkYmSSJmR0YlRiY=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= followed by the rearrangement 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

LDgqJkkiYUc2IiIiIilJInVHRiUiIiNGJkYmKihJImJHRiVGJkYoRiZJInZHRiVGJkYmKiYsKEkiZEdGJUYmKiZGK0YmSSJrR0YlRiZGJiooRilGJkYkRiZJImhHRiVGJkYmRiZGKEYmRiYqJkkiY0dGJUYmKUYsRilGJkYmKiYsKCooRilGJkY1RiZGMUYmRiYqJkYrRiZGM0YmRiZJImVHRiVGJkYmRixGJkYmKiZGJEYmKUYzRilGJkYmKiZGNUYmKUYxRilGJkYmKiZGO0YmRjFGJkYmKiZGL0YmRjNGJkYmKihGK0YmRjNGJkYxRiZGJkkiZkdGJUYm

Extract the coefficients of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= with 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

LChJImRHNiIiIiIqJkkiYkdGJEYlSSJrR0YkRiVGJSooIiIjRiVJImFHRiRGJUkiaEdGJEYlRiU=

LCgqKCIiIyIiIkkiY0c2IkYlSSJrR0YnRiVGJSomSSJiR0YnRiVJImhHRidGJUYlSSJlR0YnRiU=

then determine the values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= for which these coefficients vanish. This is done with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiNEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkkvJSlzdHJldGNoeUdGSS8lKnN5bW1ldHJpY0dGSS8lKGxhcmdlb3BHRkkvJS5tb3ZhYmxlbGltaXRzR0ZJLyUnYWNjZW50R0ZJLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGWC1GLzYlUSZzb2x2ZUYnRjJGNS1JKG1mZW5jZWRHRiQ2JC1GIzYnLUZpbjYmLUYjNistRiw2JS1GLzYlUSJFRidGMkY1LUYjNiUtRjs2JFEiMUYnRj5GMkY1RkAtRkQ2LVEiPUYnRj5GR0ZKRkxGTkZQRlJGVEZWRlktRjs2JEZCRj4tRkQ2LVEiLEYnRj5GRy9GS0Y0RkxGTkZQRlJGVC9GV1EmMC4wZW1GJy9GWlEsMC4zMzMzMzMzZW1GJy1GLDYlRmNvLUYjNiUtRjs2JFEiMkYnRj5GMkY1RkBGW3BGXnAvJStleGVjdXRhYmxlR0ZJRj5GPi8lJW9wZW5HUSJ8ZnJGJy8lJmNsb3NlR1EifGhyRidGYHAtRmluNiYtRiM2Jy1GLzYlUSJoRidGMkY1RmBwLUYvNiVRImtGJ0YyRjVGX3FGPkY+RmFxRmRxRl9xRj5GPkZfcUY+

PCQvSSJoRzYiKiYsJiomSSJiR0YlIiIiSSJlR0YlRipGKiooIiIjRipJImNHRiVGKkkiZEdGJUYqISIiRiosJiokKUYpRi1GKkYwKigiIiVGKkYuRipJImFHRiVGKkYqRjAvSSJrR0YlLCQqJiwmKiZGL0YqRilGKkYwKihGLUYqRjZGKkYrRipGKkYqRjFGMEYw

JSFH Applying these definitions of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= puts the quadratic into the form 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

LCoqJkkiYUc2IiIiIilJInVHRiUiIiNGJkYmKihJImJHRiVGJkYoRiZJInZHRiVGJkYmKiZJImNHRiVGJilGLEYpRiZGJiomLCwqJkYkRiYpSSJlR0YlRilGJiEiIiomRi5GJilJImRHRiVGKUYmRjUqKiIiJUYmRiRGJkYuRiZJImZHRiVGJkYmKihGK0YmRjRGJkY4RiZGJiomRjtGJilGK0YpRiZGNUYmLCYqJEY+RiZGNSooRjpGJkYuRiZGJEYmRiZGNUYm

The constant term, namely, 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

KiYsLComSSJhRzYiIiIiKUkiZUdGJiIiI0YnISIiKiZJImNHRiZGJylJImRHRiZGKkYnRisqKiIiJUYnRiVGJ0YtRidJImZHRiZGJ0YnKihJImJHRiZGJ0YpRidGL0YnRicqJkYyRicpRjRGKkYnRitGJywmKiRGNkYnRisqKEYxRidGLUYnRiVGJ0YnRis=

suggests defining 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

LCYqJClJImJHNiIiIiMiIiJGKCooIiIlRihJImNHRiZGKEkiYUdGJkYoISIi

and the determinant 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

LCwqKCIiIyIiIkkiYUc2IkYlKUkiZUdGJ0YkRiUhIiIqKiIiKUYlRiZGJUkiY0dGJ0YlSSJmR0YnRiVGJSoqRiRGJUkiYkdGJ0YlRilGJUkiZEdGJ0YlRiUqKEYkRiVGLkYlKUYwRiRGJUYqKihGJEYlRi1GJSlGMUYkRiVGKg==

If we define LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy as 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 via 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

LCQqKCMiIiIiIiNGJSwsKihGJkYlSSJhRzYiRiUpSSJlR0YqRiZGJSEiIioqIiIpRiVGKUYlSSJjR0YqRiVJImZHRipGJUYlKipGJkYlSSJiR0YqRiVGLEYlSSJkR0YqRiVGJSooRiZGJUYxRiUpRjNGJkYlRi0qKEYmRiVGMEYlKUY0RiZGJUYtRiUsJiokRjZGJUYlKigiIiVGJUYwRiVGKUYlRi1GLUYl

then the constant term becomes LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkMtSSNtaUdGJDYlUScmIzk2MTtGJy8lJ2l0YWxpY0dGNEYvRi8=, as we see from 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

IiIh

At this point, the quadratic can be put into the form 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

LCoqJkkiYUc2IiIiIilJInVHRiUiIiNGJkYmKihJImJHRiVGJkYoRiZJInZHRiVGJkYmKiZJImNHRiVGJilGLEYpRiZGJkkiUkdGJSEiIg==

where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= has to be used in place of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy since an expression has already been assigned to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy. To use LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy in this form of the quadratic, the assignment must be removed with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEpdW5hc3NpZ25GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Jy1JI21vR0YkNi1RIidGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRictRiw2JVEnJiM5NjE7RicvRjBGQkY+RjovJStleGVjdXRhYmxlR0ZCRj5GPkZZRj4= so that we can write 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

LCoqJkkiYUc2IiIiIilJInVHRiUiIiNGJkYmKihJImJHRiVGJkYoRiZJInZHRiVGJkYmKiZJImNHRiVGJilGLEYpRiZGJkkkcmhvR0YlISIi

The LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjdXZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn-term is removed with a rotation of the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjdXZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn-system to an LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjWFlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn-system via the equations 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 JSFH Using the abbreviations LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSRjb3NGJy9GMEY9RjktSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUScmIzk1MjtGJ0ZSRjlGOUY5Rjk= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSRzaW5GJy9GMEY9RjktSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUScmIzk1MjtGJ0ZSRjlGOUY5Rjk=, the quadratic becomes 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

LCoqJkkiYUc2IiIiIiksJiomSSJDR0YlRiZJIlhHRiVGJkYmKiZJIlNHRiVGJkkiWUdGJUYmISIiIiIjRiZGJiooSSJiR0YlRiZGKEYmLCYqJkYtRiZGK0YmRiYqJkYqRiZGLkYmRiZGJkYmKiZJImNHRiVGJilGM0YwRiZGJkkkcmhvR0YlRi8=

or equivalently, 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

LCoqJiwoKiZJImFHNiIiIiIpSSJDR0YnIiIjRihGKComSSJjR0YnRigpSSJTR0YnRitGKEYoKihJImJHRidGKEYqRihGL0YoRihGKClJIlhHRidGK0YoRigqKCwqKiZGMUYoRi5GKCEiIiomRjFGKEYpRihGKCoqRitGKEYmRihGL0YoRipGKEY3KipGK0YoRi1GKEYvRihGKkYoRihGKEkiWUdGJ0YoRjNGKEYoSSRyaG9HRidGNyomLCgqJkYmRihGLkYoRigqJkYtRihGKUYoRihGMEY3RigpRjtGK0YoRig=

The coefficients of 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 are respectively LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY2LUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEjMTJGJy9GNlEnbm9ybWFsRidGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSomY29sb25lcTtGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlgtRi82JVEpY29lZnRheWxGJ0YyRjUtSShtZmVuY2VkR0YkNiQtRiM2Ky1GLDYlRi4tRiM2JS1GOzYkUSMxMUYnRj5GMkY1RkAtRkQ2LVEiLEYnRj5GRy9GS0Y0RkxGTkZQRlJGVC9GV1EmMC4wZW1GJy9GWlEsMC4zMzMzMzMzZW1GJy1GaW42Ji1GIzYnLUYvNiVRIlhGJ0YyRjVGZG8tRi82JVEiWUYnRjJGNS8lK2V4ZWN1dGFibGVHRklGPkY+LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLUZENi1RIj1GJ0Y+RkdGSkZMRk5GUEZSRlRGVkZZLUZpbjYmLUYjNictRjs2JEZCRj5GZG9GZXFGZnBGPkY+RmhwRltxRmRvLUZpbjYmLUYjNictRjs2JFEiMkYnRj5GZG9GZXFGZnBGPkY+RmhwRltxRmZwRj5GPi1GRDYtUSI7RidGPkZHRmdvRkxGTkZQRlJGVEZob0ZZLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHRmlvLyUmZGVwdGhHRmZyLyUqbGluZWJyZWFrR1EobmV3bGluZUYnLUZENi1RIn5GJ0Y+RkdGSkZMRk5GUEZSRlRGaG8vRlpGaW8tRiw2JUYuLUYjNiUtRjs2JFEjMTNGJ0Y+RjJGNUZARkNGZW4tRmluNiQtRiM2K0Zdb0Zkb0ZccEZecUZhcUZkby1GaW42Ji1GIzYnLUY7NiRRIjFGJ0Y+RmRvRmF0RmZwRj5GPkZocEZbcUZmcEY+Rj5GXnJGYXJGXnMtRiw2JUYuLUYjNiUtRjs2JFEjMTRGJ0Y+RjJGNUZARkNGZW4tRmluNiQtRiM2K0Zdb0Zkb0ZccEZecUZhcUZkby1GaW42Ji1GIzYnRmVxRmRvRltyRmZwRj5GPkZocEZbcUZmcEY+Rj5GZnBGPg==

LCgqJkkiYUc2IiIiIilJIkNHRiUiIiNGJkYmKiZJImNHRiVGJilJIlNHRiVGKUYmRiYqKEkiYkdGJUYmRihGJkYtRiZGJg==

LCoqJkkiYkc2IiIiIilJIlNHRiUiIiNGJiEiIiomRiRGJilJIkNHRiVGKUYmRiYqKkYpRiZJImFHRiVGJkYoRiZGLUYmRioqKkYpRiZJImNHRiVGJkYoRiZGLUYmRiY=

LCgqJkkiYUc2IiIiIilJIlNHRiUiIiNGJkYmKiZJImNHRiVGJilJIkNHRiVGKUYmRiYqKEkiYkdGJUYmRi1GJkYoRiYhIiI=

The first and third of these expressions are called LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= so that the quadratic becomes LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEoY29sbGVjdEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYqLUYsNiVRKXNpbXBsaWZ5RidGL0YyLUY2NiQtRiM2Jy1JJW1zdWJHRiQ2JS1GLDYlUSJxRidGL0YyLUYjNiUtSSNtbkdGJDYkUSMxMUYnL0YzUSdub3JtYWxGJ0YvRjIvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIixGJ0ZNLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRlgvJSpzeW1tZXRyaWNHRlgvJShsYXJnZW9wR0ZYLyUubW92YWJsZWxpbWl0c0dGWC8lJ2FjY2VudEdGWC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUY2NiYtRiM2Ky1GQjYlRkQtRiM2JS1GSjYkUSMxMkYnRk1GL0YyRk8tRlM2LVEiPUYnRk1GVi9GWkZYRmVuRmduRmluRltvRl1vL0Zgb1EsMC4yNzc3Nzc4ZW1GJy9GY29GZXAtRiw2JVEiUEYnRi9GMkZSLUZCNiVGRC1GIzYlLUZKNiRRIzE0RidGTUYvRjJGT0ZgcC1GLDYlUSJRRidGL0YyLyUrZXhlY3V0YWJsZUdGWEZNRk0vJSVvcGVuR1EifGZyRicvJSZjbG9zZUdRInxockYnRmRxRk1GTUZSLUY2NiYtRiM2Jy1GLDYlUSJYRidGL0YyRlItRiw2JVEiWUYnRi9GMkZkcUZNRk0vRmdxUSJbRicvRmpxUSJdRidGUi1GLDYlUSxkaXN0cmlidXRlZEYnRi9GMkZkcUZNRk1GZHFGTQ==

LCpJJHJob0c2IiEiIiomKUkiWEdGJCIiIyIiIkkiUEdGJEYqRioqJilJIllHRiRGKUYqSSJRR0YkRipGKiooLCoqJkkiYkdGJEYqKUkiU0dGJEYpRipGJSomRjNGKilJIkNHRiRGKUYqRioqKkYpRipJImFHRiRGKkY1RipGOEYqRiUqKkYpRipJImNHRiRGKkY1RipGOEYqRipGKkYuRipGKEYqRio=

It remains to determine the value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy for which the coefficient of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJZRidGL0YyRjk= vanishes, a process aided by putting the resulting equation into the form 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 or 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 from which it follows that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtaUdGJDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZRLUYxNiVRImNGJ0Y0RjdGPkY+LUY7Ni1RIn5GJ0Y+RkBGQ0ZFRkdGSUZLRk0vRlBRJjAuMGVtRicvRlNGZW4tRjE2JVEkc2luRicvRjVGQkY+LUYsNiQtRiM2Ji1JI21uR0YkNiRRIjJGJ0Y+RlctRjE2JVEnJiM5NTI7RidGam5GPkY+Rj4tRjs2LVEiPUYnRj5GQEZDRkVGR0ZJRktGTS9GUFEsMC4yNzc3Nzc4ZW1GJy9GU0Zqby1GMTYlUSJiRidGNEY3RlctRjE2JVEkY29zRidGam5GPkZbb0Y+ and hence that 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 A Maple implementation of these last manipulations is tedious. It begins with the equation 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

LywqKiZJImJHNiIiIiIpSSJTR0YmIiIjRichIiIqJkYlRicpSSJDR0YmRipGJ0YnKipGKkYnSSJhR0YmRidGKUYnRi5GJ0YrKipGKkYnSSJjR0YmRidGKUYnRi5GJ0YnIiIh

and requires moving the terms containing LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= to the right with 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

LywmKioiIiMiIiJJImFHNiJGJkkiU0dGKEYmSSJDR0YoRiYhIiIqKkYlRiZJImNHRihGJkYpRiZGKkYmRiYsJiomSSJiR0YoRiYpRilGJUYmRiYqJkYwRiYpRipGJUYmRis=

This equation is then multiplied by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkMtSSNtbkdGJDYkUSIxRidGL0Yv, obtaining 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

LywmKioiIiMiIiJJImFHNiJGJkkiU0dGKEYmSSJDR0YoRiZGJioqRiVGJkkiY0dGKEYmRilGJkYqRiYhIiIsJiomSSJiR0YoRiYpRilGJUYmRi0qJkYwRiYpRipGJUYmRiY=

which factors to 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

LywkKioiIiMiIiJJIkNHNiJGJkkiU0dGKEYmLCZJImFHRihGJkkiY0dGKCEiIkYmRiYqJiwmKiQpRilGJUYmRi0qJClGJ0YlRiZGJkYmSSJiR0YoRiY=

Removing the abbreviations LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= leads to 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

LywkKioiIiMiIiItSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kmdGhldGFHRixGJi1JJHNpbkdGKUYtRiYsJkkiYUdGLEYmSSJjR0YsISIiRiZGJiomLCYqJClGL0YlRiZGNCokKUYnRiVGJkYmRiZJImJHRixGJg==

which can be written as 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

LyomLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJComIiIjIiIiSSZ0aGV0YUdGKUYuRi5GLiwmSSJhR0YpRi5JImNHRikhIiJGLiomLUkkY29zR0YmRipGLkkiYkdGKUYu

The expression for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkdGFuRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUkjbW5HRiQ2JFEiMkYnRjItSSNtb0dGJDYtUSJ+RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlItRiw2JVEnJiM5NTI7RidGL0YyRjJGMkYy appears after a final rearrangement and conversion via 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

Ly1JJHRhbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiIiIyIiIkkmdGhldGFHRihGLUYtKiZJImJHRihGLSwmSSJhR0YoRi1JImNHRighIiJGNA==

The angle LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy determined by this equation can be in either the first or second quadrant. When it is in the first quadrant, both LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkc2luRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRJyYjOTUyO0YnRi9GMkYyRjJGMg== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkY29zRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRJyYjOTUyO0YnRi9GMkYyRjJGMg== are positive; when it is in the second, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkY29zRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRJyYjOTUyO0YnRi9GMkYyRjJGMg== will be negative. Thus, care must be taken when determining LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy. With the coefficient of the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjWFlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn-term set to zero, the quadratic, the left side of an equation whose right side is zero, becomes 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 and hence 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 The definitions 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 and 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 complete the derivation of the formulas. JSFH

JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=