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Q.
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 $x^2+y^2+4 x+1=0$, then the maximum value of $a b$ is
Conic Sections
Solution:
Tangent on ellipse having slope 2 will be
$y=2 x \pm \sqrt{4 a^2+b^2}$
$\Theta$ It is normal to circle
$\therefore(-2,0)$ is on it.
$\Rightarrow 0=-4 \pm \sqrt{4 a^2+b^2} \Rightarrow 4 a^2+b^2=16$
$\Theta$ A.M. $\geq$ G.M. $\Rightarrow \frac{4 a ^2+ b ^2}{2} \geq 2 ab \Rightarrow 8 \geq 2 ab \Rightarrow ab \leq 4$
$\therefore$ maximum value of $a b=4$