Given, equation of circle x2+y2−2x−1=0 Coordinate of C is (1,0). Radius =√2 Let the coordinate of P be (1+√2cosθ,√2sinθ). Given equation of another circle x2+y2−2x=0 Coordinate center is (1,0) and radius =1 ∴ The two given circle are concentric. Let the circumcenter of CAB be (h,k). Circumcenter of CAB lies on mid-point of PC h=
1+√2cosθ+1
2
and k=
√2sinθ+0
2
⇒cosθ=
2h−2
√2
and sinθ=
2k
√2
⇒cos2θ+sin2θ=1 ⇒(
2(h−1)
√2
)2+(
2k
√2
)2=1 ⇒2(h−1)2+2k2=1 ⇒2h2+2−4h+2k2=1 ⇒2h2+2k2−4h+1=0 ∴ The locus of circumcenter is 2x2+2y2−4x+1=0