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Question

Let P be any moving point on the circle S1:x2+y22x1=0. A chord of contact is drawn from the point P to the circle S:x2+y22x=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

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Solution


The two circles are
S:(x1)2+y2=1
S1:(x1)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=12
CM2=(12)2
(h1)2+(k0)2=(12)2
Hence, locus of M is (x1)2+y2=12.

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