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.