JEE Main 2015 · 10 Apr · Shift 1, Mathematics question 73

Let A be $(a,0)$. The line AP, $\frac xa+\frac yh=1$, is tangent to the circle, so its distance from the origin is 4:
$$\frac{ah}{\sqrt{a^2+h^2}}=4\implies a^2=\frac{16h^2}{h^2-16}.$$
The area is $ah=\dfrac{4h^2}{\sqrt{h^2-16}}$. With $u=h^2$, $\dfrac{u^2}{u-16}$ is least at $u=32$, so $h=4\sqrt2$.