The locus of the points of intersection of 

the tangents at the extremities of the chords of the ellipse 

$x^2+2y^2=6$ which touch the ellipse $x^2+4y^2=4$ is 

We can write  $x^2+4y^2=4\text{ as }\frac{x^2}{4}+\frac{y^2}{1}=1\text{ (i) }$ 

Equation of a tangent to the ellipse$\left(i\right)$  is 

          $\frac{x}{2}\cos \theta +y\sin \theta =1$

Equation of the ellipse  $x^2+2y^2=6$  can be written as

        $\frac{x^2}{6}+\frac{y^2}{3}=1$

Suppose $\left(ii\right)$  meets the ellipse$\left(iii\right)$  at P and Q and the tan

gents at $P$ and $Q$  to the ellipse $\left(iii\right)$ intersect at $(h, k)$ , then

$\left(ii\right)$  is the chord of contact of $(h, k)$ with respect to the ellipse 

$\left(iii\right)$  and thus its equation is $\frac{hx}{6}+\frac{ky}{3}=1$               $\text{ (iv) }$

Since$\left(ii\right)$  and  $\left(iv\right)$ represent the same line 

$\begin{array}{l}\frac{h/6}{(\cos \theta )/2}=\frac{k/3}{\sin \theta }=1\\ \Rightarrow h=3\cos \theta ,k=3\sin \theta .\end{array}$     

and the locus of $(h,k)$ is $x^2+y^2=9$