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.  

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 

\[cos\theta = \frac{AO}{OB} \text{ and sin }\theta = \frac{AB}{OB}\]
\[ \Rightarrow cos\theta = \frac{x_1}{OB} \text{ and sin }\theta = \frac{y_1}{OB}\]

∴ x₁ = OBcosθ and y₁ = OBsinθ
Now, the curve is passing through (x₁, y₁)
∴ (y₁)² = 4a(x₁)
⇒( OBsinθ)² = 4a(OBcosθ) 

\[\Rightarrow {OB}^2 \sin^2 \theta = 4aO\text{ Bcos }\theta\]
\[ \Rightarrow OB = \frac{4acos\theta}{\sin^2 \theta} = 4a\text{ cosec }\theta . \cot\theta\]