Let from a point A h , k chord of contacts are drawn to the ellipse x 2 + 2 y 2 = 6 such that all these chords touch the ellipse x 2 + 4 y 2 = 4 , then locus of the point A is

Question

Let from a point A h , k chord of contacts are drawn to the ellipse x 2 + 2 y 2 = 6 such that all these chords touch the ellipse x 2 + 4 y 2 = 4 , then locus of the point A is, 4 x 2 + 9 y 2 = 36, x 2 + y 2 = 4, x 2 - y 2 = 9, x 2 + y 2 = 9

MARKS App · Accepted Answer

Let chord of contact is P Q that touches x 2 + 4 y 2 = 4 at R Now, assume R = 2 cos ⁡ θ , sin ⁡ θ Equation of P Q is h x + 2 y k = 6 … 1 T = 0 for p o i n t A h , k Again, the equation of P Q is x c o s θ 2 + y s i n θ 1 = 1 … 2 From 1 & 2 we get 2 h cos ⁡ θ = 2 k sin ⁡ θ = 6 cos ⁡ θ = h 3 , sin ⁡ θ = k 3 ⇒ x 2 + y 2 = 9

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