Let from a point A h , k chords of contact are drawn to the ellipse x 2 + 2 y 2 = 6 where all these chords touch the ellipse x 2 + 4 y 2 = 4 . Then, the perimeter (in units) of the locus of point A is

Step-by-step solution

Let the chord of contact is P Q which touches x 2 + 4 y 2 = 4 at R Now, assume R = 2 cos ⁡ θ , 2 sin ⁡ θ Equation of the chord of contact P Q is h x + 2 y k = 6 … . . i Again, the equation of the tangent P Q is x cos ⁡ θ 2 + y sin ⁡ θ 1 = 1 … . . i i From i and i i , we get, h 2 cos ⁡ θ = 2 k s i n θ = 6 cos ⁡ θ = h 3 , sin ⁡ θ = k 3 ⇒ x 2 + y 2 = 9 Hence, the perimeter of the circle is 6 π units