Find the equation of the smallest circle passing through the intersection of the line `x+y=1` and the circle `x^2+y^2=9`

Explanation

Let the given circle and line intersect at points A and B.
Equation of family of cirlces through A and B is
`x^(2)+y^(2)-9+lambda(x+y-1)=0,lambda in R`
Variable centre of the circle is `(-(lambda)/(2),-(lambda)/(2))`.
The smallest circle of this family is that for which A and B are end points of diameter.
So, centre `(-(lambda)/(2),-(lambda)/(2))`. lies ont he chord `x+y-1=0`.
`implies -(lambda)/(2),-(lambda)/(2)-1=0`
`implies lambda= -1`
Using this value for `lambda`, the equation of the smallest circle is
`x^(2)+y^(2)-9-(x+y-1)=0`
or `x^(2)+y^(2)-x-y-8=0`