Question

For the given parabola find the coordinates of focus, axis, the equation of the directrix, and the length of the latus rectum.
\[{{x}^{2}}=-9y\]

Answer 1

Hint: Compare the equation of the parabola \[{{x}^{2}}=-9y\] with the standard equation of the parabola and get the value of a. The axis of the parabola \[{{x}^{2}}=-4ay\] is the y-axis. We know that for the parabola \[{{x}^{2}}=-4ay\] , the coordinate of the focus is \[\left( 0,-a \right)\] , the equation of the directrix is \[y=a\] , the length of the latus rectum is 4a. Now, using the value of \[a\] get the coordinates of focus, axis, the equation of the directrix, and the length of the latus rectum.

Complete step-by-step answer:
According to the question, we have a parabola and we have to find the coordinates of focus, axis, the equation of the directrix, and the length of the latus rectum.
The equation of the parabola, \[{{x}^{2}}=-9y\] ……………………………(1)

The given equation of the parabola is on the form of the standard equation of the parabola \[{{x}^{2}}=-4ay\] ………………………(2)
Now, on comparing equation (1) and equation (2), we get
\[\begin{align}
  & \Rightarrow -4ay=-9y \\
 & \Rightarrow 4ay=9y \\
 & \Rightarrow 4a=9 \\
\end{align}\]
\[\Rightarrow a=\dfrac{9}{4}\] ………………………….(3)
We know that for the parabola \[{{x}^{2}}=-4ay\] , the coordinate of the focus is \[\left( 0,-a \right)\] ………………………(4)
From equation (3), we have the value of a, \[a=\dfrac{9}{4}\] .
Now, putting the value of \[a\] in equation (4), we get
The coordinate of the focus is \[\left( 0,-\dfrac{9}{4} \right)\] ………………………………….(5)
We also know that for the parabola \[{{x}^{2}}=-4ay\] , the equation of the directrix is \[y=a\] ………………………..(6)
From equation (3), we have the value of a, \[a=\dfrac{9}{4}\] .
Now, putting the value of \[a\] in equation (6), we get
The equation of the directrix is \[y=\dfrac{9}{4}\] ………………………………………….(7)
We know the formula for the length of latus rectum, Latus rectum = \[4a\] …………………………………..(8)
From equation (3), we have the value of a, \[a=\dfrac{9}{4}\] .
Now, putting the value of \[a\] in equation (8), we get
The length of latus rectum, Latus rectum = \[4a=4\times \dfrac{9}{4}=9\] …………………………………….(10)
We know that the axis of the parabola \[{{x}^{2}}=-4ay\] is the y-axis.
Since the parabola \[{{x}^{2}}=-9y\] is of the form \[{{x}^{2}}=-4ay\] so, the axis of the parabola \[{{x}^{2}}=-9y\] is also the y-axis and the equation of the y-axis is \[x=0\] …………………………..(12)
Now, from equation (5), equation (7), equation (10), and equation (11), we have
The coordinate of the focus is \[\left( 0,-\dfrac{9}{4} \right)\] .
The equation of the directrix is \[y=\dfrac{9}{4}\] .
The length of the latus rectum, Latus rectum = 9.
The axis of the parabola is the y-axis i.e, \[x=0\] .

Note: In this question, one might compare the equation of the parabola \[{{x}^{2}}=-9y\] with the standard form that is, \[{{y}^{2}}=4ax\] . This is wrong because the equation of the parabola \[{{x}^{2}}=-9y\] is not in the form of the equation of the parabola, \[{{y}^{2}}=4ax\] . The given equation of the parabola \[{{x}^{2}}=-9y\] is in the form of the standard equation of the parabola \[{{x}^{2}}=-4ay\] . So, we have to compare the equations \[{{x}^{2}}=-9y\] and \[{{x}^{2}}=-4ay\] .