Question

The latus rectum of a parabola whose focal chord is $PSQ$ such that $SP = 3$ and $SQ = 2$is given by
$A)\dfrac{{24}}{5}$
$B)\dfrac{{12}}{5}$
$C)\dfrac{6}{5}$
$D)$None of the above

Answer 1

First we have to define what the terms we need to solve the problem are.
Let the parabola be defined as a curve where any of the points is at the distance from the focal points or the fixed points and also from the straight fixed lines (the directrix). Latus rectum can be defined as the chord that will be passing through the focus and from the perpendicular axis to the given directrix from the parabola.
Formula used: Semi latus rectum $\dfrac{{2SP.SQ}}{{SP + SQ}}$

Complete step by step answer:
Since from the question the semi latus is the harmonic means which can be yielded by SP and SQ points where P and Q are the extremities that will be the focal chord and also S is the focal point. Since semi latus of the given rectum of the parabola is the harmonic mean of the length that points will focus to the parabola given; hence the formula is $2a = \dfrac{{2({L_1}{L_2})}}{{{L_1} + {L_2}}}$
Hence the semi latus of the rectum from the given parabola is the HM of the segments of a focal chord,
Thus, using the formula of the Semi latus rectum =$\dfrac{{2SP.SQ}}{{SP + SQ}}$ and such that $SP = 3$ and $SQ = 2$
Substitute the given values in the formula we get, and further solving we get the semi latus rectum as $\dfrac{{12}}{5}$ and we need to find the latus rectum and hence multiply with two.

So, the correct answer is “Option A”.

Note: since the option$B)\dfrac{{12}}{5}$ is more likely to be correct but only if they ask about the semi latus rectum. Hence for this option, all other options are eliminated and A is the only correct option.
The harmonic mean of the given length can be taken into two parts like SP and SQ and that will have a focal chord equal to the length of the semi latus rectum.