# The Point of Parabola Closed to Focus is the Vertex

The point of parabola closed to focus is the vertex. Let the given parabola be

The vertex and the focus of the parabola are $O\left( {0,0} \right)$ and $F\left( {a,0} \right)$ respectively, as shown in the given diagram. Let $P\left( {x,y} \right)$ be any point on the parabola, then its distance from the focus is

Putting the value of ${y^2}$ from equation (i) and equation (ii), we have

The distance of the vertex from the focus is

It is also clear from the above diagram that for any point $P\left( {x,y} \right)$ of the parabola,

This shows that the distance of the vertex from the focus is less than the distance of $P$ from the focus, so the point of the parabola closed to focus is the vertex.