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(C) The first region is the area between the circle $x^{2} + y^{2} = a^{2}$ and the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$. The area

Vedclass · Accepted Answer

$12$

Vedclass · Answer

(C) The first region is the area between the circle $x^{2} + y^{2} = a^{2}$ and the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$. The area is $\pi a^{2} - \pi ab = 30\pi$,which simplifies to $a^{2} - ab = 30$. The second region is the area between the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ and the circle $x^{2} + y^{2} = b^{2}$. The area is $\pi ab - \pi b^{2} = 18\pi$,which simplifies to $ab - b^{2} = 18$. Adding these two equations: $(a^{2} - ab) + (ab - b^{2}) = 30 + 18$,so $a^{2} - b^{2} = 48$. Subtracting the second from the first: $(a^{2} - ab) - (ab - b^{2}) = 30 - 18$,which gives $a^{2} - 2ab + b^{2} = 12$. Thus,$(a - b)^{2} = 12$.

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