Question

If a tangent of slope 2 on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is normal to the circle $x^2+y^2+4x+1=0$, then the maximum value of $ab$ is

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (D)

Tangent on ellipse having slope 2 will be
$y=2x\pm \sqrt{4a^2+b^2}$
$\Theta$ It is normal to circle
$\therefore (-2,0)$ is on it.
$\Rightarrow 0=-4\pm \sqrt{4a^2+b^2}\Rightarrow 4a^2+b^2=16$
$\Theta$ A.M. $\ge$ G.M. $\Rightarrow \frac{4a^2+b^2}{2}\ge 2ab\Rightarrow 8\ge 2ab\Rightarrow ab\le 4$
$\therefore$ maximum value of $ab=4$