A square is inscribed inside the ellipse x^2/a^2 + y^2/b^2 = 1 , the length of the side of the square is

Q: A square is inscribed inside the ellipse $\large \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ , the length of the side of the square is

(A) $\frac{ab}{\sqrt{a^2 + b^2}}$

(B) $\frac{2ab}{\sqrt{a^2 + b^2}}$

(C) $ \sqrt{a^2 + b^2} $

(D) None of these

Sol. Ellipse

Let one vertex of the square be (p, p) then

$\large \frac{p^2}{a^2} + \frac{p^2}{b^2} = 1$

$\large  p^2 = \frac{a^2 b^2}{a^2 + b^2} $

⇒ $\large  p = \frac{ab}{\sqrt{a^2 + b^2}}$

⇒  Length of side $\large  = \frac{2ab}{\sqrt{a^2 + b^2}}$