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Let P be any moving point on the circle ...

Let `P` be any moving point on the circle `x^2+y^2-2x=1. A B` be the chord of contact of this point w.r.t. the circle `x^2+y^2-2x=0` . The locus of the circumcenter of triangle `C A B(C` being the center of the circle) is `2x^2+2y^2-4x+1=0` `x^2+y^2-4x+2=0` `x^2+y^2-4x+1=0` `2x^2+2y^2-4x+3=0`

Text Solution

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The two circles are
`(x-1)^(2)+y^(2)=1` (1)
`(x-1)^(2)+y^(2)=2` (2)
Clearly, the second circle is the director circle of the first circle.
`:. /_ AB =90^(@)`
S, APBC is square.
Now, circumcentre of the right-angled triangle CAB will be midpoint of hypotenuse.
Let midpoint of AB be M(h,k).
Now, `CM=CB sin 45^(@) =(1)/(sqrt(2))`
So, `(h-1)^(2)+k^(2)=(1)/(2)`
Hence, locus of M is `(x-1)^(2)+y^(2)=(1)/(2)`
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