The minimum area of triangle formed by the tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $ and coordinate axes is

Q: The minimum area of triangle formed by the tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $ and coordinate axes is

(A) ab sq. units

(B) $\frac{a^2 + b^2}{2} $ sq. units

(C) $\frac{(a + b)^2}{2} $ sq. units

(D) $\frac{a^2 + ab + b^2}{3} $ sq. units

Sol. A tangent of the given ellipse is $\large y = m x + \sqrt{a^2 m^2 + b^2}$

It meets the axes at $\large (\frac{-\sqrt{a^2 m^2 + b^2}}{m} , 0 ) $ and $(0 , \sqrt{a^2 m^2 + b^2})$

Hence the area of the triangle is $\large \frac{1}{2}|\frac{a^2 m^2 + b^2}{m}| $

$\large = \frac{1}{2}|a^2 m + \frac{b^2}{m}| \ge a b$