Question

If a circle passes through the point $(a,b)$ and cuts the circle $x^2+y^2=4$ orthogonally, then the locus of its center is:

• A. $2ax$ $+2by+(a^2+b^2+4)=0$
• B. $2ax$$+2by-(a^2+b^2+4)=0$
• C. $2ax-2by$ $+(a^2+b^2+4)=0$
• D. $2ax-2by$ $-(a^2+b^2+4)=0$

Answer 1

Answer: (B)

$2ax$$+2by-(a^2+b^2+4)=0$

Let circle be $S:x^2+y^2+2gx+2fy+c=0$

Center $(-g,-f)$

$S$ passes through $(a,b)$

$\Rightarrow x^2+y^2+2gx+2fy+c=0-\left(1\right)$

$S$ and $x^2+y^2=0$ are orthogonal.

$2g{g}^{\prime }+2f{f}^{\prime }=c+c^{\prime }$

$\Rightarrow 0+0=c-4$

$\therefore c=4$

$(-g,-f)=(x,y)$

$\left(1\right)\Rightarrow a^2+b^2-2ax-2yb+4=0$

$\therefore 2ax+2by-(a^2+b^2+4)=0$