Question

The equation of the circle concentric with the circle $x^2+y^2-6x+12y+15=0$ and of double its area is

• A. $x^2+y^2-6x+12y-15=0$
• B. $x^2+y^2-6x+12y-30=0$
• C. $x^2+y^2-6x+12y-25=0$
• D. $x^2+y^2-6x+12y-20=0$

Answer 1

Answer: (A)

The equation of the circle is
$S\equiv x^2+y^2-6x+12y+15=0$
Let the equation of concentric circle of given circles, is
$S_2=x^2+y^2-6x+12y+15=0$
On comparing the circle $S_1$ with,
$x^2+y^2+2gx+2fy+c=0$
$\Rightarrow g=-3,f=6,c=15$
Then, radius of circle is
$=\sqrt{g^2+f^2-c}$
$=\sqrt{9+36-15}$
$=\sqrt{45-15}=\sqrt{30}$ units
and centre is $(-g,-f)=(3,-6)$
Now, the area of the circle $S$ is $=\pi$ (radius) ${}^2$
$=\pi (\sqrt{30}{)}^2$
$=30\pi$
Let the radius of the concentric circle is $r_2$.
$r_2=\sqrt{g^2+f^2-c}$
$=\sqrt{9+36-c}$
$=\sqrt{45-c}$
Then, according to question, the area of concentric circle
$=2\times$ area of $S$
$=2\times 30\pi =60\pi$
$\Rightarrow \pi r_2^2=60\pi$
$\Rightarrow (\sqrt{45-c}{)}^2=60$
$\Rightarrow 45-c=60$
$\Rightarrow c=-15$
Hence, the equation of concentric circle is
$x^2+y^2+2(-3)x+2(6)y+(-15)=0$
$x^2+y^2-6x+12y-15=0$