Question

If the area of the circle $4x^2+4y^2+8x-16y+\lambda =0$ is $9\pi$ sq. units, then the value of $\lambda$ is

• A. $4$
• B. $-4$
• C. $16$
• D. $-16$
• E. $-8$

Answer 1

The correct option is D $-16$

Given, equation of circle is
$4x^2+4y^2+8x-16y+\lambda =0$
$x^2+y^2+2x-4y+\frac{\lambda }{4}=\mathrm{0...}\left(i\right)$
Here, $g=1;f=-2;c=\frac{\lambda }{4}$
$\therefore$ Radius $=\sqrt{g^2+f^2-c}\Rightarrow r=\sqrt{1+4-\frac{\lambda }{4}}=\sqrt{5-\frac{\lambda }{4}}$
$\therefore$ Area $=9\pi$
$\therefore \pi {\left(\sqrt{5-\frac{\lambda }{4}}\right)}^2=9\pi \Rightarrow 5-\frac{\lambda }{4}=9$
$\Rightarrow \lambda =-16$