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.

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