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 $\frac{2h}{\cos \theta }=\frac{2k}{\sin \theta }=6$
$\cos \theta =\frac{h}{3},\sin \theta =\frac{k}{3}\Rightarrow x^2+y^2=9$