For real numbers $a, b\left(a>b>0\right)$, let

Area $\left\{\left(x, y\right):x^2+y^2\le a^2\right.$ and $\left.\frac{x^2}{a^2}+\frac{y^2}{b^2}\ge 1\right\}=30\pi$

and

Area $\left\{\left(x, y\right):x^2+y^2\ge b^2\right.$ and $\left.\frac{x^2}{a^2}+\frac{y^2}{b^2}\le 1\right\}=18\pi$

Then the value of $\left(a-b\right){}^2$ is equal to _____.

We know that area of circle $={\mathrm{\pi r}}^2$ ($r$ is radius of circle) and area of ellipse $=\mathrm{\pi ab}$ (where $a \& b$ are length of semi major and semi minor axis)

Now given area between circle and ellipse  $\Rightarrow \pi a^2-\pi ab=30\pi$  .....(i)

And $\Rightarrow \pi ab-\pi b^2=18\pi$ ....(ii)

On subtracting equation (i) - (ii), we get ${\left(a-b\right)}^2=a^2-2ab+b^2=12$.