Let $\mathrm{E}$ be an ellipse whose major axis is $\mathrm{X}$-axis and minor axis is $\mathrm{Y}$-axis. If…
Let $\mathrm{E}$ be an ellipse whose major axis is $\mathrm{X}$-axis and minor axis is $\mathrm{Y}$-axis. If the distance of a point $\left(\frac{5}{2}, 2 \sqrt{3}\right)$ on $\mathrm{E}$ from its foci are $\frac{7}{2}$ and $\frac{13}{2}$, then the eccentricity of the ellipse $\mathrm{E}$ is
$3 / 5$
$1 / 5$
$1 / \sqrt{5}$
$1 / \sqrt{2}$
Solution
Let the point $\left(\frac{5}{2}, 2 \sqrt{3}\right)$ be $\mathrm{P}$
Then, $S P+S P=2 a$
$\Rightarrow \frac{7}{2}+\frac{13}{2}=2 a \Rightarrow a=5$
The focus of the ellipse are $( \pm a e, 0) \equiv( \pm 5 e, 0)$
$\begin{aligned}
& \mathrm{SP}=\sqrt{\left(5 e-\frac{5}{2}\right)^2+(2 \sqrt{3}-0)^2} \\
& \Rightarrow\left(\frac{7}{2}\right)^2=\left(5 e-\frac{5}{2}\right)^2+12 \\
& \Rightarrow \frac{49}{4}-12=\left(5 e-\frac{5}{2}\right)^2 \Rightarrow \frac{6}{4}\left(5 e-\frac{5}{2}\right)^2 \\
& \Rightarrow 5 e-\frac{5}{2}=\frac{1}{2} \Rightarrow 5 e=3 \Rightarrow e=\frac{3}{5}
\end{aligned}$