Question

If $PSQ$ and $P{S}^{\prime }R$ are two chords of an ellipse through its foci $S$ and $S^{\prime }$ , then $\frac{PS}{SQ},\frac{P{S}^{\prime }}{S^{\prime }R}=$

• A. $2\left(\frac{1+e^2}{1-e^2}\right)$
• B. $\frac{1+e^2}{1-e^2}$
• C. $3\left(\frac{1-e^2}{1+e^2}\right)$
• D. $\frac{1-e^2}{1+e^2}$

Answer 1

Answer: (A)

$2\left(\frac{1+e^2}{1-e^2}\right)$

Refer to the figure.

Polar form of the equation of ellipse is given by,

$\frac{l}{r}=1+ecos\theta$

For segment SP, we can write,

$\frac{l}{SP}=1+ecos\theta$ (1)

For segment SQ, we can write,

$\frac{l}{SQ}=1+ecos(\pi +\theta )$

$\therefore \frac{l}{SQ}=1-ecos\theta$ (2)

Adding (1) and (2), we get,

$\frac{l}{SP}+\frac{l}{SQ}=1+ecos\theta +1-ecos\theta$

$\therefore \frac{l}{SP}+\frac{l}{SQ}=2$

Multiply both sides by $SP$, we get,

$l+l\frac{SP}{SQ}=2\times SP$ (3)

Similarly, for chord $P{S}^{{}^{\prime }}R$, we can write,

$\therefore l+l\frac{P{S}^{{}^{\prime }}}{S^{{}^{\prime }}R}=2\times P{S}^{{}^{\prime }}$ (4)

Adding equations (3) and (4), we get,

$l+l\frac{SP}{SQ}+l+l\frac{P{S}^{{}^{\prime }}}{S^{{}^{\prime }}R}=(2\times SP)+(2\times P{S}^{{}^{\prime }})$

$\therefore 2l+l(\frac{SP}{SQ}+\frac{P{S}^{{}^{\prime }}}{S^{{}^{\prime }}R})=2(PS+P{S}^{{}^{\prime }})$

By the property of ellipse, $PS+P{S}^{{}^{\prime }}=2a$

$\therefore 2l+l(\frac{SP}{SQ}+\frac{P{S}^{{}^{\prime }}}{S^{{}^{\prime }}R})=2\left(2a\right)$

$\therefore l(\frac{SP}{SQ}+\frac{P{S}^{{}^{\prime }}}{S^{{}^{\prime }}R})=2\left(2a\right)-2l$

$\therefore l(\frac{SP}{SQ}+\frac{P{S}^{{}^{\prime }}}{S^{{}^{\prime }}R})=2(2a-l)$

$\therefore (\frac{SP}{SQ}+\frac{P{S}^{{}^{\prime }}}{S^{{}^{\prime }}R})=2(\frac{2a}{l}-1)$

Now, $l$ is y coordinate of end of latus rectum which is given by,

$l=\frac{b^2}{a}$

$\therefore (\frac{SP}{SQ}+\frac{P{S}^{{}^{\prime }}}{S^{{}^{\prime }}R})=2(\frac{2a}{\left(\frac{b^2}{a}\right)}-1)$

$\therefore (\frac{SP}{SQ}+\frac{P{S}^{{}^{\prime }}}{S^{{}^{\prime }}R})=2(\frac{2a^2}{b^2}-1)$

Now, $\frac{b^2}{a^2}=1-e^2$

$\therefore b^2=a^2(1-e^2)$

$\therefore (\frac{SP}{SQ}+\frac{P{S}^{{}^{\prime }}}{S^{{}^{\prime }}R})=2(\frac{2a^2}{a^2(1-e^2)}-1)$

$\therefore (\frac{SP}{SQ}+\frac{P{S}^{{}^{\prime }}}{S^{{}^{\prime }}R})=2(\frac{2}{(1-e^2)}-1)$

$\therefore (\frac{SP}{SQ}+\frac{P{S}^{{}^{\prime }}}{S^{{}^{\prime }}R})=2\left(\frac{2-1+e^2}{(1-e^2)}\right)$

$\therefore (\frac{SP}{SQ}+\frac{P{S}^{{}^{\prime }}}{S^{{}^{\prime }}R})=2\left(\frac{1+e^2}{1-e^2}\right)$