Question

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

• A. $ab$
• B. $2ab$
• C. $4ab$
• D. $a^2{b}^2$

Answer 1

The correct option is A $ab$

Let equation of tangent be $\frac{x\cos \varphi }{a}+\frac{y\sin \varphi }{b}=1$

Area of Triangle formed $=\frac{1}{2}\frac{a}{\cos \varphi }\times \frac{b}{\sin \varphi }$

$=\frac{ab}{\sin 2\varphi }$

$\sin 2\varphi$ has maximum value $1.$

$\therefore$ Minimum area $=ab$