Ellipse Question 1
Question 1 - 2024 (01 Feb Shift 2)
Let $P$ be a point on the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$. Let the line passing through $P$ and parallel to $y$-axis meet the circle $x^{2}+y^{2}=9$ at point $Q$ such that $P$ and $Q$ are on the same side of the $x$-axis. Then, the eccentricity of the locus of the point $R$ on $P Q$ such that $P R: R Q=4: 3$ as $P$ moves on the ellipse, is :
(1) $\frac{11}{19}$
(2) $\frac{13}{21}$
(3) $\frac{\sqrt{139}}{23}$
(4) $\frac{\sqrt{13}}{7}$
Show Answer
Answer (4)
Solution
$P(3 \cos \theta, 2 \sin \theta)$
$Q(3 \cos \theta, 3 \sin \theta)$
$h=3 \cos \theta$
$k=\frac{18}{7} \sin \theta$
$\therefore$ locus $=\frac{x^{2}}{9}+\frac{49 y^{2}}{324}=1$
$e=\sqrt{1-\frac{324}{49 \times 9}}=\frac{\sqrt{117}}{21}=\frac{\sqrt{13}}{7}$
