If any tangent to the ellipse 25 x 2 + 9 y 2 = 225 meets the coordinate axes at A and B such that O A = O B then, the length A B is equal to (where, O is the origin)

Step-by-step solution

Given, equation of the ellipse is x 2 9 + y 2 25 = 1 Let, a point on it be P 3 cos ⁡ θ , 5 sin ⁡ θ and equation of the tangent at P is x 3 cos ⁡ θ + y 5 sin ⁡ θ = 1 which meets the x -axis at A 3 c o s θ , 0 and y -axis at B 0 , 5 s i n θ Let, O A = O B = λ cos ⁡ θ = 3 λ ,   sin ⁡ θ = 5 λ Now, c o s 2 θ + s i n 2 θ = 1 9 λ 2 + 25 λ 2 = 1 ⇒ λ 2 = 3 4 ∴ λ = 34 Now, A B = 2 λ =