The equation of radical axis of two circles is `x + y = 1`. One of the circles has the ends ofa diameter at the points `(1, -3) and (4, 1)` and the other passes through the point (1, 2).Find the equations of these circles.

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`