Question

Let the tangents drawn to the circle, $x^2+y^2=16$ from the point $P(0,h)$ meet the $x-$axis at points $A$ and $B$. If the area of $\Delta APB$ is minimum, then h is equal to :

• A. $3\sqrt{2}$
• B. $4\sqrt{3}$
• C. $3\sqrt{3}$
• D. $4\sqrt{2}$

Answer 1

The correct option is D $4\sqrt{2}$

Given circle is $x^2+y^2=16$ and tangents are drawn from $P(0,h)$ such that they intersect x-axis at $A$ and $B$
Area of $\Delta APB$ is minimum, only when it is a right angled triangle with right angle at $P$.
$\therefore$ Equations of AP and BP are $x+y-h=0$ and $x-y+h=0$ respectively
As AP is tangent to the circle distance from origin to $x+y-h=0$ is equal to radius.
$\Rightarrow \frac{h}{\sqrt{2}}=4$
$\therefore h=4\sqrt{2}$
Hence, option D.