TS EAMCET 18-July-2022 Shift 1 Solved Paper

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