Let P be any moving point on the circle x2+y2−2x=1 AB be the chord of contact of this point w.r.t. the circle x2+y2−2x=0 The locus of the circumcenter of triangle CAB (C being the center of the circle) is

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

1. A

$2{\mathrm{x}}^{2}+2{\mathrm{y}}^{2}-4\mathrm{x}+1=0$

2. B

${\mathrm{x}}^{2}+{\mathrm{y}}^{2}-4\mathrm{x}+2=0$

3. C

${\mathrm{x}}^{2}+{\mathrm{y}}^{2}-4\mathrm{x}+1=0$

4. D

$2{\mathrm{x}}^{2}+2{\mathrm{y}}^{2}-4\mathrm{x}+3=0$

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### Solution:

Let P be
The circumcenter of triangle ABC is the midpoint of PC.
Therefore,

or $\left[2\left(\mathrm{h}-1\right){\right]}^{2}+\left(2\mathrm{k}{\right)}^{2}=2$
or $2\left(\mathrm{h}-1{\right)}^{2}+{\mathrm{k}}^{2}-1=0$
or $2{\mathrm{x}}^{2}+2{\mathrm{y}}^{2}-4\mathrm{x}+1=0$  