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, $SP+SP=6$.
If chord $PQ$ subtends $90$ angle at centre of ellipse, then locus of $A$ is -

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

Answer 1

Answer: (B)

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

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