Question

If a tangent at a point P on a parabola makes angle a with its focal distance, then angle between the tangent and axis of the parabola is:

• A. $a$
• B. $\frac{a}{2}$
• C. $2a$
• D. ${90}^{\circ }$

Answer 1

The correct option is A $a$

Let $T$ be the tangent.

Foot of perpendicular at directrix is $M$.

We know that tangent bisects angle between perpendicular to directrix from $P$ and focal distance of $P$.

$=>\angle MPB=\angle BPS=a$

Now, $△MPB$is right angle triangle$=>\angle MPB=90-a$

Now, $△OBA$is also right angle triangle$=>\angle OAB=a$

Now,$\angle OAB=\angle PAS($vertically opposite angle$)$

$\angle PAS=a$.