The point of parabola closed to focus is the vertex.
Let the given parabola be
The vertex and the focus of the parabola are and respectively as shown in the given diagram. Let be any point on the parabola, then its distance from focus is
Putting the value of from equation (i) and equation (ii), we have
The distance of vertex from focus is
It is also clear from the above diagram that for any point of the parabola,
This shows that the distance of the vertex from the focus is less than the distance of from the focus, so the point of parabola closed to focus is the vertex.