Question

Let the focus $S$ of the parabola $y^{2}=8x$ lie on the focal chord $PQ$ of the same parabola. If the length $QS=3$ units, then the ratio of length $PQ$ to the length of the latus rectum of the parabola is

• A. $\frac{2}{\sqrt{5}}$
• B. $\frac{4}{5}$
• C. $\frac{5}{4}$
• D. $\frac{9}{8}$

Answer 1

Answer: (D)

For the focal chord $PQ$ , we know that,
$PS,4,SQ$ are in H.P.
$\Rightarrow \frac{1}{P S},\frac{1}{4},\frac{1}{3}$ are in A.P.
$\Rightarrow \frac{1}{P S}=\frac{2}{4}-\frac{1}{3}=\frac{1}{6}\Rightarrow PS=6$
$\Rightarrow PQ=6+3=9$
$\Rightarrow \frac{P Q}{length \, of \, l a t u s \, re c t u m}=\frac{9}{8}$