F=x²+y² is the equation of a circle with centre at the origin and radius=√F.
When this circle has a tangent given by x+2y=15, which can be written y=15/2-x/2. The gradient is -1/2 so the gradient of the perpendicular radius is 2 and the equation of the radius is y=2x.
The tangent point is given by 2x=15/2-x/2, 5x/2=15/2 and x=3, making y=6. So the tangent point is (3,6) and F=9+36=45. The equation of the circle is x²+y²=45, making 45 the minimum value of F.