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. $\dfrac{{48}}{5}$

Answer 1

Hint:Here, we are going to use the property of parabola for the latus rectum of the parabola.
The Latus rectum of a parabola is a line segment perpendicular to the axis of the parabola, throughout the focus and whose endpoints lie in the parabola.

Complete step by step solution:
Given that: SP $ = 3$
            SQ $ = 2$
By using the property of parabola-
As we know that - The harmonic mean of the lengths of two parts into which the focus divides the focal chord is equal to the length of semi-latus rectum.
$\dfrac{{2SP \times SQ}}{{SP + SQ}} = \dfrac{1}{2}$
Now, substituting the values of SP & SQ
$\begin{array}{l}
\therefore L = \dfrac{{4 \times 3 \times 2}}{{3 + 2}}\\
\therefore L = \dfrac{{24}}{5}
\end{array}$
Latus rectum, $l = \dfrac{{24}}{5}$ is the required answer.

Therefore, the given option A is correct.

Note: Always remember all the properties of parabola.

Additional Information: The parabola is symmetric about its axis and the axis is perpendicular to the directrix. The vertex always passes through the vertex and the focus. The tangent at the vertex is parallel to the given directrix.