$\text{ A circle passes through the focus of parabola }{x}^2=4y\text{ . If the circle touches the parabola at }$$\text{ (6,9), then square of its radius is }$

Equation of tangent to parabola x² = 4y at point P(6,9) is: 

$6\mathrm{x}=2(\mathrm{y}+9)\text{ or }3\mathrm{x}-\mathrm{y}-9=0$

Equation of family of circles touching tangent at point P is: 

${(x-6)}^2+{(y-9)}^2+\lambda (3x-y-9)=0$

Circle passes through the point (0,1) =focus of the parabola. 

$\therefore \lambda =10$

Thus, the equation of circle is:

$\begin{array}{l}{x}^2+y^2+18x-28y+27=0\\ \text{ Radius of the circle is }5\sqrt{10}\text{ . }\end{array}$