The parabola has the y-axis as its axis of symmetry and the point lies on the y-axis.
If (a,a²) is the closest point on the curve to (0,5) then the distance from the point to the curve is:
√((5-a)²+a²)=√(25-9a²+a⁴)=√(25-81/4+81/4-9a²+a⁴)=√(19/4+(9/2-a²)²).
The minimum value of the square root is when a²=9/2, a=±3√2/2.
The y values for x=a are 9/2 in each case, so the closest points are (-3√2/2,9/2) and (3√2/2,9/2).