Mathematics: A parabola has a focus of (6,–6) and a directrix of y = –2. Which of the following could be the equation of the parabola?

A parabola has a focus of (6,–6) and a directrix

dyamondgeorge1oy4nex

28.01.2022

$-8(y+4) =(x-6)^{2}$

Step-by-step explanation:

The standard form of a parabola is given by the following equation:

$(x-h)^{2} =4p(y-k)$

Where the focus is given by:

F(h,k+p)

The vertex is:

V=(h,k)

And the directrix is:

y-k+p=0

Now, using the previous equations and the information provided by the problem, let's find the equation of the parabola.

If the focus is (-6,6):

F=(h,k+p)=(6,-6)

Hence:

$h=6\\\\k+p=-6\hspace{10}(1)$

And if the directrix is y=-2 :

$-2-k+p=0\\\\k-p=-2\hspace{10}(2)$

Using (1) and (2) we can build a 2x2 system of equations:

$k+p=-6\hspace{10}(1)\\k-p=-2\hspace{10}(2)$

Using elimination method:

(1)+(2)

$k+p+k-p=-6+(-2)\\\\2k=-8\\\\k=-\frac{8}{2}=-4\hspace{10}(3)$

Replacing (3) into (1):

$-4+p=-6\\\\p=-6+4\\\\p=-2$

Therefore:

$(x-6)^{2} =4(-2)(y-(-4)) \\\\(x-6)^{2} =-8(y+4)$

So, the correct answer is:

Option 3

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