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Answer

`x^(2)+y^(2)-(15)/(2)x-(y)/(2)+(7)/(2)=0`

Explanation

Equations of the circle having the end points of diameter as `(1,-3)` and (4,1) is
`(x-1)(x-4)+(y+3)(y-1)=0`
`:. x^(2)+y^(2)-5x+2y+1=0`
Equation of common chord is `x+y-1=0`
Equation of family of circles through point of intersection common chord and circle is
`x^(2)+y^(2)-5x+2y+1+lambda(x+y-1)=0`.

If it passes through (1,2) , then `lambda =-(5)/(2)`
Therefore, the equation of circle is `x^(2)+y^(2)-(15)/(2)x-(y)/(2)+(7)/(2)=0`