Question

lf a circle passes through the point $(a,b)$ and cuts the circle $x^2+y^2=r^2$ orthogonally then the locus of its centre is

• A. a circle
• B. a parabola
• C. an ellipse
• D. a straightline

Answer 1

Answer: (D)

a straightline

$r_1^2+r_2^2=(c_1{c}_2{)}^2$
$r^2+g^2+f-c=g^2+f^2$
$r^2=c$
So, $x^2+y^2+2gx+2fy+c=0$
$(a,b)$ on circle,
So, $a^2+b^2+2ag+2fb+r^2=0$

So, locus of centre $(-g,-f)$ is
$-x\left(2a\right)-\left(2a\right)y+a^2+b^2+r^2=0$
$\Rightarrow$ striaght line.