6 Functional Relations for Ellipse.

Article 6
Functional Relations for Ellipse.

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The functional relations that exist between the sectorial measure and each of its characteristic ratios are the same for all elliptic, including circular, sectors (Art. 4). Let $P_{1}, P_{2}$ be corresponding points on an ellipse and a circle, referred to the conjugate directions $O_{1} A_{1}, O_{1} B_{1}$ and $O_{2} A_{2}, O_{2} B_{2}$ , the latter pair being at right angles; let the angle $A_{2} O_{2} P_{2} = θ$ in radian measure; then

\begin{matrix} \frac{S_{2}}{K_{2}} = \frac{\frac{1}{2} a_{2}^{2} θ}{\frac{1}{2} a_{2}^{2}} = θ . & (6) \\ ∴ \frac{x_{2}}{a_{2}} = cos \frac{S_{2}}{K_{2}}, \frac{y_{2}}{b_{2}} = sin \frac{S_{2}}{K_{2}}; [a_{2} = b_{2} \end{matrix}

hence, in the ellipse, by Art. 3,

\begin{matrix} \frac{x_{1}}{a_{1}} = cos \frac{S_{1}}{K_{1}}, \frac{y_{1}}{b_{1}} = sin \frac{S_{1}}{K_{1}} . & (7) \end{matrix}

Prob. 4.: Given $x_{1} = \frac{1}{2} a_{1}$ ; ﬁnd the measure of the elliptic sector $A_{1} O_{1} P_{1}$ . Also ﬁnd its area when $a_{1} = 4, b_{1} = 3, ω = 6 0^{\circ}$ .
Prob. 5.: Find the characteristic ratios of an elliptic sector whose measure is $\frac{1}{4} π$ .
Prob. 6.: Write down the relation between an elliptic sector and its triangle. (See Art. 5.)

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Article 6Functional Relations for Ellipse.

Article 6
Functional Relations for Ellipse.