The given equation of the curve is
y2 = 4ax ... (1)
Differentiating both sides of (1) with respect to x, we get
2y
= 4a ; ⇒
=
=
... (2)
If Ψ be the angle which the tangent to the curve at (x, y) makes with the positive direction of x-axis then tan Ψ =
or tan Ψ =
... (3) . [using (2)]
At x = a, then from (1),
y2 = 4a , a =
4a2 ⇒ y = ± 2a
Hence, we get two points (a, 2a) and (a, - 2a) on the curve.
At (a, 2a) x = a, y = 2a and let Ψ =
Ψ1 ∴ from (3), tan
Ψ1 =
= 1 = tan 45°
⇒
Ψ1 = 45°
At (a, -2a), x = a, y = -2a and let Ψ =
Ψ1 ∴ from (3), tan
Ψ2 =
= - 1 = tan 135 °
or
Ψ2 = 135°
Hence the required angle between tangents
to (1) at (a, 2a) and (a, -2a) =
Ψ1 -
Ψ2 = 135°- 45° = 90°.
This shows that the tangent lines to (1) at (a, 2a) and (a, -2a) are perpendicular to each other.