Question

Maximum area of a circle centered at the origin, which is inscribed in the parabola $y=x^2-100$, can be expressed as $a\pi$, then the value of $\left[\sqrt{a}\right]$ is
(where [.] denotes the greatest integer function)

Answer 1

Maximum area of circle holds when circle is tangent to parabola.

$x^2+y^2=r^2$, $y=x^2-100$
$y^2+y+100-r^2=0$
$D=0$

$1-4(100-r^2)=0\Rightarrow r^2=\frac{399}{4}$

Area $=\pi r^2=\pi \times \frac{399}{4}$

$a=\frac{399}{4}$
So $\left[\sqrt{a}\right]=9$