Question

If the locus of the centre of the variable circle which intersects the circles $x^2+y^2=4$ and $x^2+y^2+2x+4y=6$ orthogonally has the equation $C_{1}x+C_{2}y=1$, then the value of $\left(C_1+C_2\right)$ is

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (C)

Let the required circle be $x^2+y^2+2gx+2fy+c=0$
It is orthogonal with $x^2+y^2=4$
$\therefore c-4=0$
$\therefore c=4$
$\therefore$ and also with $x^2+y^2+2x+4y-6=0$
$\therefore 2(1\cdot g+2f)=4-6$
$\Rightarrow 2(g+2f)=-2$
$\Rightarrow g+2f=-1$
$\therefore$ locus of the centre is
$-x-2y=-1$
$\Rightarrow x+2y-1=0$
$\Rightarrow C_1=1$ and $C_2=2$
Hence $C_1+C_2=3$