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`

Options

A. `x^(2)+y^(2)+x+y-8=0`
B. `x^(2)+y^(2)-x-y-8=0`
C. `x^(2)+y^(2)-x+y-8=0`
D. none of these

Answer

B

Explanation

The equation of a circle passing through the intersection of the given line and the given circle is

`x^(2)|y^(2)|9|lambda(x+y-1)=0 " " ...(i)`
This circle will be smallest if x+y=1 is a diameter of it.
`:. -(lambda)/(2)-(lambda)/(2)=1 rArr lambda=-1`
Putting `lambda=-1` in (i), we get `x^(2)+y^(2)-x-y-8=0` as the equation of the required circle.