Question

The latus rectum of the parabola$y^2=4ax$, whose focal chord is $PSQ$, such that $SP=3$ and $SQ=2$ are given by

• A. $\frac{24}{5}$
• B. $\frac{12}{5}$
• C. $\frac{6}{5}$
• D. $\frac{1}{5}$

Answer 1

The correct option is A

$\frac{24}{5}$

Explanation for the correct option:

Latus-rectum and semi length of focal chord:

A chord through the focus of the parabola parallel to the directrix of the parabola is called the latus rectum of the parabola

The half of the length of this chord is called semi latus rectum

The semi latus rectum is a harmonic mean between the two segments of a focal chord

Let $l$ be the length of the semi latus rectum of the parabola

We know,

$\frac{2}{l}=\frac{1}{SP}+\frac{1}{SQ}$

$\Rightarrow$$\frac{2}{l}=\frac{1}{3}+\frac{1}{2}=\frac{5}{6}$

$\Rightarrow$ $l=\frac{12}{5}$

$\Rightarrow$ $2l=\frac{24}{5}$

The length of the latus rectum $=\frac{24}{5}$.

Hence, option A is correct.