Question

If the equation of parabola is ${x^2} = - 9y$ then the equation of the directrix and the length of latus rectum is
A. $y = - \dfrac{9}{4},8$
B. $y = - \dfrac{9}{4},9$
C. $y = \dfrac{9}{4},9$
D.None of these

Answer 1

Hint : We are provided with an equation of the parabola. By comparing the equation to the general equation of parabola, we come to know that the equation is of the shape downwards U shape. The direction of the directrix is given by the equation $y = a$ and the length of the latus rectum by the $L = 4a$

Complete step-by-step answer :
In the question, we are given an equation of parabola. ${x^2} = - 9y$
The general equation for a parabola is ${x^2} = - 4ay$ where a is positive. The equation is given in the x- quadratic terms so the focus will belong on the y-axis and the negative sign tells the direction which should be parabola in the downwards direction.
Given equation of parabola is ${x^2} = - 9y$ $\ldots \left( 1 \right)$
And the general equation is ${x^2} = - 4ay$ $\ldots \left( 2 \right)$
Comparing the above marked equations
$4a = 9 \\\ \Rightarrow a = \dfrac{9}{4} \;$
For the general equation of the parabola ${x^2} = - 4ay$ , the equation of directrix is $y = a$ . So, the given equation ${x^2} = - 9y$ , the equation of directrix is $y = \dfrac{9}{4}$
And the length of latus rectum is
$L = 4a \\\ = 4\left( {\dfrac{9}{4}} \right) \\\ = 9 \;$
Hence, the required equation of the directrix is $y = \dfrac{9}{4}$ and the length of the latus rectum is $9$ .
So, the correct option is C.
So, the correct answer is “Option C”.

Note : The equation of the parabola tells the direction of the directrix. When the equation is of the form ${x^2} = - 4ay$ then the direction of the directrix is parallel to the y-axis. The latus rectum is the chord that passes through the focus and is perpendicular to the axis of the parabola.