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 

detailed solution

Correct option is C

We can write  x2+4y2=4 as x24+y21=1 (i)  Equation of a tangent to the ellipse (i)  is           x2cos⁡θ+ysin⁡θ=1Equation of the ellipse  x2+2y2=6  can be written as        x26+y23=1Suppose (ii)  meets the ellipse (iii)  at P and Q and the tangents at P  and Q  to the ellipse (iii) intersect at (h, k) , then (ii)  is the chord of contact of (h, k) with respect to the ellipse  (iii)  and thus its equation is hx6+ky3=1                (iv) Since (ii)  and  (iv)  represent the same line h/6(cos⁡θ)/2=k/3sin⁡θ=1⇒ h=3cos⁡θ,k=3sin⁡θ.     and the locus of (h,k)  is  x2+y2=9