Equation of the circle of minimum radius which touches both the parabolas $y=x^2+2x+4$and $x=y^2+2y+4$is

Given parabolas are symmetrical about the line y = x so they have a common normal with slope -1 it meets the parabolas at   $\left(\frac{-1}{2},\frac{13}{4}\right),\left(\frac{13}{4},\frac{-1}{2}\right)$  hence the required circle is $x^2+y^2-\frac{11}{4}x-\frac{11}{4}y-\frac{13}{4}=0$