Question

If the parabola ${{y}^{2}}=4ax$ passes through the point $\left( 2,-6 \right)$, then the length of the latus rectum is
(a) 9
(b) 16
(c) 18
(d) $\dfrac{9}{2}$

Answer 1

Hint: In the equation of the parabola, the variable a is still not given and thus can be found by using the fact that as the point passes through the given point, the coordinates of the point should satisfy the equation of the parabola. Then, using the formula for latus rectum of a parabola, we can obtain the answer to the given question.

Complete step-by-step answer:

We are given that the equation of the parabola is ${{y}^{2}}=4ax$. Also, it is given that the point (2, -6) passes through the parabola. Therefore, its coordinates x=2 and y=-6 should satisfy the equation of the parabola. Therefore, putting x=2 and y=-6 in the equation of the parabola, we get

${{\left( -6 \right)}^{2}}=4a\times 2\Rightarrow a=\dfrac{-6\times -6}{4\times 2}=\dfrac{9}{2}........(1.1)$

Therefore, using the value of a, the equation of the parabola becomes

${{y}^{2}}=4\times \dfrac{9}{2}\times x\Rightarrow {{y}^{2}}=18x..........(1.2)$

Now, the formula for the latus rectum of the parabola is given by

Length of latus rectum of the parabola ${{y}^{2}}=4ax$ = 4a

Therefore, using the equation of the given parabola from equation (1.1), we obtain

Length of the latus rectum of the parabola \[{{y}^{2}}=4\times \dfrac{9}{2}\times x\] is equal to $4a=4\times \dfrac{9}{2}=18$.

This matches the option (c) given in the question. Thus, option (c) is the correct answer to this question.

Note: Note that as the length of the latus rectum is 4a and the equation of the parabola is given by ${{y}^{2}}=4ax$, we could have written the latus rectum as the coefficient of x in the RHS from equation (1.2) itself.