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Question

Find the equation of the circle passing through point of intersection of the circle x2+y28x2y+7=0 and x2+y24x+10y+8=0 and its center lie on y-axis.

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Solution

The equation of a circle passing through the point of intersection of the given circles is
x2+y28x2y+7+4(x2+y24x+10y+15)=0
Where h is as arbitary constant
or (1+k)x2+(1+k)y24x(2+k)2y(15k)+7+8k=0
or x2+y24(2+k1+k)x2(15k1+k)y+7+8k1+k=0
where k1
The coordinates of center of circle is {2(2+k)(1+k)(15k)(1+k)}
Since center of the circle lies on y axis.
Abscissa =0
2(2+k1+k)=0k=2
From (1) equation of required circle
x2+y28x2y+2(x2+y24x+10y+8)=0
x2+y2+22y+9=0

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