Question

Let P be any moving point on the circle $x^2+y^2-2x=1$. AB be the chord of contact of this point w.r.t. the circle $x^2+y^2-2x=0$. The locus of the circumcentre of the triangle CAB (C being center of the circles) is

• A. $2x^2+2y^2-4x+1=0$
• B. $x^2+y^2-4x+2=0$
• C. $x^2+y^2-4x+1=0$
• D. $2x^2+2y^2-4x+3=0$

Answer 1

Answer: (A)

$2x^2+2y^2-4x+1=0$

Let P be $(1+\sqrt{2}\cos \theta ,\sqrt{2}\sin \theta )$ and C is $(1,0)$.
Circumcentre of triangle $ABC$ is midpoint of $PC$.
$\Rightarrow 2h=1+\sqrt{2}\cos \theta +1$
and $2k=\sqrt{2}\sin \theta$
$\Rightarrow \left[2\right(h-1){]}^2+(2k{)}^2=2$
$\Rightarrow 2(h-1{)}^2+k^2-1=0$
$\Rightarrow 2x^2+2y^2-4x+1=0$