Question

A parabola can be drawn given a focus of (-6,3) and a directrix of x = -4. Write
the equation of the parabola in any form.

Answer 1

Answer:

$(y-3)^2=-4(x+5)$

Step-by-step explanation:

Given the directrix is vertical, the parabola must have a horizontal axis since it's perpendicular to the directrix.

Therefore, we can use the form $(y-k)^2=4p(x-h)$ where the focus is $(h+p,k)$ and the directrix is $x=h-p$.

Therefore:

$h+p=-6$

$h-p=-4$

$2h=-10$

$h=-5$

So:

$x=h-p$

$-4=-5-p$

$1=-p$

$-1=p$

This means that the parabola opens to the left since $p<0$. Remember that $p$ describes the distance between the vertex and focus point.

We can tell that $k=3$ given the focus point

In conclusion, the equation of the parabola is $(y-3)^2=-4(x+5)$