Question

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. [NCERT EXEMPLAR]

Answer 1

Let the coordinates of the point on the parabola be B (x₁, y₁).
Let BO be the line segment
In right triangle AOB
$$\begin{gathered} \mathrm{cos\theta }=\frac{\mathrm{AO}}{\mathrm{OB}}\mathrm{and}\mathrm{sin\theta }=\frac{\mathrm{AB}}{\mathrm{OB}} \\ \Rightarrow \mathrm{cos\theta }=\frac{x_1}{\mathrm{OB}}\mathrm{and}\mathrm{sin\theta }=\frac{y_1}{\mathrm{OB}} \end{gathered}$$
∴ x₁ = OBcosθ and y₁ = OBsinθ
Now, the curve is passing through (x₁, y₁)
∴ (y₁)² = 4a(x₁)
⇒( OBsinθ)² = 4a(OBcosθ)
$$\begin{gathered} \Rightarrow {\mathrm{OB}}^2{\sin }^2\mathrm{\theta }=4a\mathrm{OBcos\theta } \\ \Rightarrow \mathrm{OB}=\frac{4a\mathrm{cos\theta }}{{\sin }^2\mathrm{\theta }}=4a\mathrm{cosec}\theta .\cot \theta \end{gathered}$$