about to only mathematics

Explanation

Let AB be the length of chord intercepted by the circle on `y+x=0`.
Let CM be perpendicular to BA from center C(h,k).

Also, `y-x=0` and `y+x=0` are perpendicule to each other. Therefore , OPCM is a rectangle. So,
`CM=OP=4 sqrt(2)`
Let r be the radius of the circle.
Therefore, in `Delta CAM`,
`AC^(2)=AM^(2)+MC^(2)`
or `r^(2)=(3sqrt(2))^(2)+(4sqrt())^(2)`
or `r^(2)=(5sqrt(2))^(2)`
or `r=5 sqrt(2)`
Also, the coordinates of P are
`(0-4sqrt(2) cos 45^(@),0-4sqrt(2)sin 45^(@))` or `(-4,-4)`
The slpe of PC is -1 and CP `=5 sqrt(2)`.
Therefore, the coordinates of C are
`(-4+5sqrt(2)cos 135^(@),-4+5 sqrt(2) sin 135^(@))` or `(-9,1)`
Hence, equation of the circle is
`(x+9)^(2)+(y-1)^(2)=50`
or `x^(2)+y^(2)+18x-2y+32=0`