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.