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

{y^2} = 4ax\,\,\,\,{\text{ - - - }}\left( {\text{i}} \right)

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


parabola-closed-focus

|PF| = \sqrt {{{\left( {x - a} \right)}^2} + {{\left( {y - 0} \right)}^2}} = \sqrt {{{\left( {x - a} \right)}^2} + {y^2}} \,\,\,\,{\text{ - - - }}\left( {{\text{ii}}} \right)

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

|PF| = \sqrt {{{\left( {x - a} \right)}^2} + 4ax} = \sqrt {{{\left( {x + a} \right)}^2}} = x + a\,\,\,\,{\text{ - - - }}\left( {{\text{iii}}} \right)

The distance of the vertex from the focus is

|OF| = a\,\,\,\,{\text{ - - - }}\left( {{\text{iv}}} \right)

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

\begin{gathered} x > 0 \Rightarrow x + a > a \\ \Rightarrow |PF| > |OF| \\ |OF| > |PF| \\ \end{gathered}

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.