Question

Prove that the midpoint between focus and directrix of a parabola lies on the parabola?

Answer 1

Please see below.

Explanation:

1. It is well known that the closest distance between a point and a line is the perpendicular from the point to the line.

2. Parabola is the locus of a point that moves so that its distance from a given point called focus and a given line called directrix is always equal.

Now if we draw a perpendicular from focus on to directrix , this is the shortest distance between focus and directrix (from (1) above).

Hence, midpoint of this perpendicular (this is also the vertex), which is also on parabola as it is equidistant from vertex and directrix, is the closest point to the focus on the parabola.

graph{(20(y-1)+(x+3)^2)((x+3)^2+(y-1)^2-0.08)(y-6)((x+3)^2+(y+4)^2-0.08)=0 [-20, 20, -10, 10]}