Find the length of the line segment joining the vertex of the parabola y² = 4ax and a point on the parabola where the line - segment makes an angle θ to the x - axis.

Given that we need to find the length of the line joining the vertex of parabola y² = 4ax and a point on the parabola where the line segment makes an angle θ to the x - axis.

We know that vertex of the parabola is (0, 0).

We know that the equation of the line passing through origin and making angle θ to the x - axis is given by y = (tanθ)x.

Substituting y value in the equation of parabola we get,

⇒ (xtanθ)² = 4ax

⇒ x²tan²θ = 4ax

⇒ xtan²θ = 4a

⇒

⇒ y = tanθx

⇒

⇒

The point on the parabola is .

We know that the distance between the two points (x₁, y₁) and (x₂, y₂) is .

⇒

⇒

⇒

⇒

⇒

⇒

⇒

⇒ S = 4a²cotθcosecθ.

∴The distance is 4a²cotθcosecθ.