# Find the equation of the parabola that satisfies the following conditions: Focus (0, –3); directrix y = 3

Question:

Find the equation of the parabola that satisfies the following conditions: Focus $(0,-3)$; directrix $y=3$

Solution:

Focus $=(0,-3) ;$ directrix $y=3$

Since the focus lies on the $y$-axis, the $y$-axis is the axis of the parabola.

Therefore, the equation of the parabola is either of the form $x^{2}=4 a y$ or

$x^{2}=-4 a y$

It is also seen that the directrix, $y=3$ is above the $x$-axis, while the focus

$(0,-3)$ is below the $x$-axis. Hence, the parabola is of the form $x^{2}=-4$ ay.

Here, $a=3$

Thus, the equation of the parabola is $x^{2}=-12 y$.