Let P be any moving point on the circle S1:x2+y2−2x−1=0. A chord of contact is drawn from the point P to the circle S:x2+y2−2x=0. If C is the centre and A,B are the points of contact of circle S, then the locus of the circumcentre of △CAB is
A
(x−1)2+y2=12
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B
x2+(y−1)2=12
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C
(x−1)2+y2=1
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D
x2+(y−1)2=1√2
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Solution
The correct option is A(x−1)2+y2=12
The two circles are S:(x−1)2+y2=1 S1:(x−1)2+y2=2 Radius of S is 1 and radius of S1 is √2 with both S and S1 having centre as C(1,0). S1 is the director circle of S. ∴∠APB=π2 Also, ∠ACB=π2
Now, circumcentre of the right angled isosceles triangle △CAB would lie on the mid-point of AB. So, let the point be M≡(h,k) Now, CM=CBsin45∘=1√2 ⇒CM2=(1√2)2 ⇒(h−1)2+(k−0)2=(1√2)2 Hence, locus of M is (x−1)2+y2=12.