Question

Area common to curve $y=\sqrt{9-x^2}$ and $x^2+y^2=6x$ is

• A. $\frac{\pi +\sqrt{3}}{4}$ sq. units
• B. $\frac{\pi -\sqrt{3}}{4}$ sq. units
• C. $3\left(\pi +\frac{\sqrt{3}}{4}\right)$ sq. units
• D. $3\left(\pi -\frac{3\sqrt{3}}{4}\right)$ sq. units

Answer 1

Answer: (D)

$x^2+y^2=9$, a circle with centre $\left(0,0\right)$ and radius $3\dots \left(i\right)$
$x^2+y^2-6x=0$, a circle with centre $\left(3,0\right)$ and radius $3\dots \left(ii\right)$
On solving $\left(i\right)$ and $\left(ii\right)$, we get
$x=\frac{3}{2},y^2=9-\frac{9}{4}=\frac{27}{4}$
$\Rightarrow y=\frac{3\sqrt{3}}{2}$

$\therefore A=2\underset{3/2}{\overset{3}{\int }}\left(\sqrt{9-x^2}\right)dx$
$=3\left(\pi -\frac{3\sqrt{3}}{4}\right)$ sq. units