Question

As shown in the figure above, $AB$ is an arc of a circle with center at $O$. Point $A$ lies on the curve $y=x^2-b$, where $b$ is a constant. Find the value of $b$ if the area of the shaded region $AOB$ is $\pi$.

Answer 1

$A$ lies on the curve $y=x^2-b$ as well as on the line $y=0$

$\therefore$ the coordinates of $A$ are $(-\sqrt{b},0)$ (Obtained after equating $x^2-b$ to $0$)

Since $AB$ is an arc of a circle with centre $O,OA$ is the radius.

$\therefore$ radius $=\sqrt{b}$

Area of quarter circle becomes $\frac{1}{4}\times \pi r^2=\pi$

$\therefore \frac{1}{4}\times b=1$

$\therefore b=4$