Question

How can we show that the middle point on perpendicular from focus to directrix lies on the parabola?

Answer 1

Please see below.

Explanation:

We know that

1. Parabola is the locus of a point which moves so that its distance from a point called focus and from a line called directrix is always equal.
2. The closest distance from focus to directrix is perpendicular from focus on to directrix .
3. Hence middle point of this perpendicular would lie on the parabola , as its distance from focus as well as directrix is same. This point is called vertex.
4. As such, it would also be the closest point to directrix.