Find the equation of auxiliary circle of conic which passes through (1,1) and is having foci (4,5) and (2,3)

Question

Please answer my question i am not getting it from so long

Answer 1

 

 So the center is at the midpoint or (3,4)

 

The distance of the radius is from (3,4) to (1,1)

 

 sqrt ( (3-1)^2 + (4-1)^2) = sqrt( 2^2 + 3^2) = sqrt ( 4+9) = sqrt(13)

 

 So the equation of the circle is (x-3)^2 + (y-4)^2 = 13