Question

An ellipse whose distance between foci $S$ and $S^{\prime}$ is 4 units is inscribed in the triangle $ABC$ touching the sides $AB , AC$ and $BC$ at $P , Q$ and $R$. If centre of ellipse is at origin and major axis along $x$-axis, $SP + S P =6$.
If chord $PQ$ subtends $90$ angle at centre of ellipse, then locus of $A$ is -

• A. $25 x^2+81 y^2=620$
• B. $25 x^2+81 y^2=630$
• C. $9 x^2+16 y^2=25$
• D. none of these

Answer 1

Answer: (B)

$2 a e=4$
$2 a =6$
$e =2 / 3$
$b ^2= a ^2\left(1-e^2\right)$
$=9\left(1-\frac{4}{9}\right)$
$=5$

Let $A$ be $( h , k )$ then chord of contact $PQ$ is
$\frac{h x}{9}+\frac{k y}{5}=1$
Homogenizing the equation of ellipse
$\frac{x^2}{9}+\frac{y^2}{5}=\left(\frac{h x}{9}+\frac{k y}{5}\right)^2 $
$x^2\left(\frac{h^2}{81}-\frac{1}{9}\right)+y^2\left(\frac{k^2}{25}-\frac{1}{5}\right)+\frac{2 h k}{45} x y=0 $
$\text { coefficient of } x^2+\text { coefficient of } y^2=0 $
$\frac{h^2}{81}-\frac{1}{9}+\frac{k^2}{25}-\frac{1}{5}=0 \Rightarrow 25 x^2+81 y^2=630$