25.10: Appendix 25C Analytic Geometric Properties of Ellipses

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Consider Equation (25.3.20), and for now take \(\varepsilon<1\), so that the equation is that of an ellipse. We shall now show that we can write it as

\[\frac{\left(x-x_{0}\right)^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \nonumber \]

where the ellipse has axes parallel to the x - and y -coordinate axes, center at (\(x_{0}\), 0) , semi-major axis a and semi-minor axis b . We begin by rewriting Equation (25.3.20) as

\[x^{2}-\frac{2 \varepsilon r_{0}}{1-\varepsilon^{2}} x+\frac{y^{2}}{1-\varepsilon^{2}}=\frac{r_{0}^{2}}{1-\varepsilon^{2}} \nonumber \]

We next complete the square,

\[\begin{array}{l}
x^{2}-\frac{2 \varepsilon r_{0}}{1-\varepsilon^{2}} x+\frac{\varepsilon^{2} r_{0}^{2}}{\left(1-\varepsilon^{2}\right)^{2}}+\frac{y^{2}}{1-\varepsilon^{2}}=\frac{r_{0}^{2}}{1-\varepsilon^{2}}+\frac{\varepsilon^{2} r_{0}^{2}}{\left(1-\varepsilon^{2}\right)^{2}} \Rightarrow \\
\left(x-\frac{\varepsilon r_{0}}{1-\varepsilon^{2}}\right)^{2}+\frac{y^{2}}{1-\varepsilon^{2}}=\frac{r_{0}^{2}}{\left(1-\varepsilon^{2}\right)^{2}} \Rightarrow \\
\frac{\left(x-\frac{\varepsilon r_{0}}{1-\varepsilon^{2}}\right)^{2}}{\left(r_{0} /\left(1-\varepsilon^{2}\right)\right)^{2}}+\frac{y^{2}}{\left(r_{0} / \sqrt{1-\varepsilon^{2}}\right)^{2}}=1
\end{array} \nonumber \]

The last expression in (25.C.3) is the equation of an ellipse with semi-major axis

\[a=\frac{r_{0}}{1-\varepsilon^{2}} \nonumber \]

semi-minor axis

\[b=\frac{r_{0}}{\sqrt{1-\varepsilon^{2}}}=a \sqrt{1-\varepsilon^{2}} \nonumber \]

and center at

\[x_{0}=\frac{\varepsilon r_{0}}{\left(1-\varepsilon^{2}\right)}=\varepsilon a \nonumber \]

as found in Equation (25.B.10).