one circle has radius 5 and its centre (0,5). A second circle has radius of 12 and its centre (12,0). What is the length of the radius of a third circle which passes through the centre of the second circle and both the points of intersection of the first two circles.

For first circle centre is (0 , 5) and radius is 5 so equation of the circle,

(*x* - 0)^{2 }+ (*y* - 5)^{2} = 5^{2} *x*^{2 }+ *y*^{2 }- 10y = 0

For second circle centre is (12, 0) and radius is 12 so,

(*x* - 12)^{2 }+ (*y *- 0)^{2} = 12^{2} *x*^{2 }+ *y*^{2 }- 24*x* = 0

Now third circle is passing through these two circles intersection points so it will be

*x*^{2 }+ *y*^{2 }- 10y + λ(*x*^{2 }+ *y* ^{2 }- 24*x*) = 0

Also third circle is passing through point (12,0) so putting this point in above equationwe get *λ* = 1 and put this value we get equation of third circle,*x*^{2 }+* y*^{2 }- 12*x* + 5*y *= 0.

This gives centre (6 , 5/2) and radius = 6.5

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