Discussion Forum



1)

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


A) $a^{2}+b^{2}$

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

C) ab

D) $\frac{(a-b)^{2}}{2}$

Answer:

Option C

Explanation:

Equation of tangent at $(a \cos \theta,b \sin \theta)$ to the ellipse is 

$\frac{x}{a} \cos \theta+\frac{y}{b} \sin \theta=1$

 6102021770_h1.PNG

 

Coordinates of P and Q are

   $\left(\frac{a}{\cos \theta},0\right)$  and   $\left(0,\frac{b}{\sin \theta}\right)$, respectively 

 Area of $\triangle OPQ$=$\frac{1}{2}|\frac{a}{\cos \theta} \times \frac{b}{\sin \theta}|=\frac{ab}{|\sin 2 \theta|}$

  $\therefore$   Minimum area =ab