`Aa n dB` are two points in the xy-plane, which are `2sqrt(2)` units distance apart and subtend an angle of `90^0` at the point `C(1,2)` on the line `x-y+1=0` , which is larger than any angle subtended by the line segment `A B` at any other point on the line. Find the equation(s) of the circle through the points `A ,Ba n dCdot`

Explanation

AB subtends the greatest angle at C, SO, theline `x-y+1=0` touches the circle at C and, hence, AB is the diameter.
The family of circles touching the line
`x-y+1=0` at point `(1,2)` is `(x-1)^(2)+(y-2)^(2)+lambda(x-y+1)=0`
Its radius is
`sqrt(((lambda-2)/(2))^(2)+((lambda+4)/(2))^(2)-(5+lambda))=sqrt(2)`
or `lambda=+-2`
Therefore, the equations of the circles are
`x^(2)y^(2)-6y+7=0`
and `x^(2)+y^(2)-4x-2y+3=0`