Question

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

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

Answer 1

The correct option is B $2ax+2by-(a^2+b^2+p^2)=0$

Let the centre be $(\alpha ,\beta )$

also it cuts the circle at $x^2+y^2=p^2$ orthogonally

So, $2(-\alpha )-0+2(-\beta )\times 0=c_1-p^2$

$\Rightarrow c_1=p^2$

$\because$ the $e{q}^n$ of circle is $x^2+y^2-2\alpha x+2\beta y+p^2=0$

it passes through $(a,b)\Rightarrow a^2+b^2-2\alpha a-2\beta b+p^2=0$

locus $\Rightarrow 2ax+2by-(a^2+b^2+p^2)=0$

So, option (B) is correct.