Question

If tangent line at the point p on a parabola makes an angle alpha with the focal distance then angle between the tangent and axis of parabola is

Answer 1

The angle between the tangent and axis of parabola is α.

Parabola is y² = 4ax & S is focus

∴OS=a

We know that co−ordinates of P is (at², 2at)

Also Focal distance = |x−coordinate|+a

∴PS = at²+a

As per the given question, SPQ = α

Equation of tangent at (x1, y1)

yy1 = 2a (x+x1)

Set (x1,y1) = (at², 2at) to get the tangent at P

y.2at = 2a (x+at²)

∴yt = x+at²

For Q, y=0

0 = x+at²

∴x = −at²

Thus, OQ = at²

⇒QS = OS+OQ

⇒a+at²

⇒SQ = PS

⇒∠SQP = ∠SPQ

⇒∠SQP = α