Question

Let from a point $A\left(h , k\right)$ chord of contacts are drawn to the ellipse $x^{2}+2y^{2}=6$ such that all these chords touch the ellipse $x^{2}+4y^{2}=4$ , then locus of the point $A$ is

• A. $4x^{2}+9y^{2}=36$
• B. $x^{2}+y^{2}=4$
• C. $x^{2}-y^{2}=9$
• D. $x^{2}+y^{2}=9$

Answer 1

Answer: (D)

Let chord of contact is $PQ$ that touches $x^{2}+4y^{2}=4$ at $R$


Now, assume $R=\left(2 \cos \theta , \sin ⁡ \theta \right)$
Equation of $PQ$ is $hx+2yk=6$ … $\left(1\right)$
$(T = 0)$ for point A (h , k)
Again, the equation of $PQ$ is
$\frac{x \cos \theta }{2}+\frac{y \sin \theta }{1}=1$ … $\left(2\right)$
From $\left(1\right)$ &$\left(2\right)$ we get $\frac{2 h}{\cos \theta }=\frac{2 k}{\sin ⁡ \theta }=6$
$\cos \theta =\frac{h}{3},\sin ⁡ \theta =\frac{k}{3}\Rightarrow x^{2}+y^{2}=9$