Question

Prove that the length of the chord of a parabola which passes through the focus and which is inclined at 30° to the axis of the parabola is four times the length of the latus rectum.

Answer 1

Solution:

Equation of parabola in polar form is

$\begin{aligned} &\frac{2 \mathrm{a}}{\mathrm{SP}}=1-\cos 30^{\circ} \\ &\Rightarrow \mathrm{SP}=\frac{2 \mathrm{a}}{1-\cos 30^{\circ}} \end{aligned}$

$\frac{2 \mathrm{a}}{\mathrm{r}}=1-\cos \theta$

Let $\mathrm{PP}^{\prime}$  be the focal chord then parametric angles of P and P' are $30^{\circ}$ and $\pi+ 30^{\circ}$  respectively then

Also 

$\frac{2 \mathrm{a}}{\mathrm{SP}^{\prime}}=1-\cos \left(\pi+30^{\circ}\right) \\\Rightarrow \mathrm{SP}^{\prime}=\frac{2 \mathrm{a}}{1-\cos \left(\pi+30^{\circ}\right)}$

Length of focal chord = PS + PS'

$\begin{aligned} &\Rightarrow 1=\frac{2 \mathrm{a}}{1-\cos 30^{\circ}}+\frac{2 \mathrm{a}}{1-\cos \left(\pi+30^{\circ}\right)} \\ &\Rightarrow 1=\frac{2 \mathrm{a}}{1-\cos 30^{\circ}}+\frac{2 \mathrm{a}}{1+\cos 30^{\circ}} \end{aligned}$

$\begin{aligned} &\Rightarrow 1=2 \mathrm{a}\left\{\frac{1+\cos 30^{\circ}+1-\cos 30^{\circ}}{1-\cos ^{2} 30^{\circ}}\right\} \\ &\Rightarrow 1=2 \mathrm{a}\left\{\frac{2}{1-\frac{3}{4}}\right\} \\ &\Rightarrow 1=16 \mathrm{a} \\ &\Rightarrow 1=4 \times 4 \mathrm{a} \\ &\Rightarrow 1=4 \times \text { length of latus rectum } \\ &\text { Hence proved. } \end{aligned}$