Find the coordinates of the focus, axis of the parabola ,the equation of directrix and the length of the latus rectum for $x^{2} = - 9 y$

Question

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Answer 1

The given equation is $x^{2} = - 9 y$
Here the coefficient of $y$ is negative.
Hence, the parabola opens downwards
On comparing this equation with $x^{2} = - 4 a y$ , we get
$- 4 a = - 9$
$\Rightarrow a = \frac{9}{4}$
$∴$ Coordinates of the focus $= (0, - a) = (0, - \frac{9}{4})$
Axis of the parabola is the $y$ -axis i.e $x = 0$
Equation of directrix $y = a$ i.e. $y = \frac{9}{4}$
Length of latus rectum $= 4 a = 9$

Answer 2

The given equation is

x^{2} = - 9 y

Here the coefficient of

y

is negative.
Hence, the parabola opens downwards
On comparing this equation with

x^{2} = - 4 a y

, we get

- 4 a = - 9

\Rightarrow a = \frac{9}{4}

∴

Coordinates of the focus

= (0, - a) = (0, - \frac{9}{4})

Axis of the parabola is the

y

-axis i.e

x = 0

Equation of directrix

y = a

i.e.

y = \frac{9}{4}

Length of latus rectum

= 4 a = 9