Question

Is the following statement true?
The foot of perpendicular (H) from the focus (S) on any tangent to a parabola at any point P lies on the tangent at vertex.

Answer 1

Hint: We have to assume parabola which bisect the angle between the focal chord through P and perpendicular from P and perpendicular from P on the directrix.

Complete step-by-step answer:
Without loss of generality, Let’s assume the parabola is
$x^2=4ay$
The focus is (0,a) and the slope at any point $\left(c,{\displaystyle \frac{c^2}{4a}}\right)$ is ${\displaystyle \frac{c}{2a}}$ and the tangent equation is
$y={\displaystyle \frac{c^2}{4a}}={\displaystyle \frac{c}{2a}}\left(x-c\right)$
Let the distance d be
$d={\displaystyle \frac{4a\left(a\right)-2c\left(0\right)-c^2+2c^2}{\sqrt{16a^2+4c^2}}}$

Now let’s find its maximum
 $$d={\displaystyle \frac{4a^2+c^2}{\sqrt{16a^2+4c^2}}}$$
$d={\displaystyle \frac{1}{2}}\sqrt{4a^2+c^2}$
This distance has its maximum varying value of c at c=0
So d=a
Now we can say that perpendicular drawn from focus on any tangent to a parabola at any point lies on the tangent at vertex.

NOTE:
Whenever you come to this type of problem assume such a point on parabola which is mentioned above. By using this we can easily get the result that the foot of perpendicular (H) from the focus (S) on any tangent to a parabola at any point P lies on the tangent at vertex.