Question

Can someone explain to me how to do this?
Find the standard form of the equation of the parabola with a focus at (-4, 0) and a directrix at x = 4.

Answer 1

This is very easy to do. All you have to do is take points (-4,0) and (4, y) and insert them into the formula:

\((x - x_1)^2 + (y - y_1)^2 = (x - x_2)^2 + (y - y_2)^2\)

Upon doing so, you get:
\((x - (-4))^2 + (y - 0)^2 = (x - 4)^2 + (y - y)^2\)

Answer 2

Okay, now simplify that to get:
\((x + 4)^2 + y^2 = (x - 4)^2\)

Answer 3

Then expand the squares to get:

\(x^2 + 8x + 16 + y^2 = x^2 - 8x + 16\)

Answer 4

Notice that \(x^2\) and \(16\) cancel:

\(8x + y^2 = -8x\)

Answer 5

so the answer should be y^2=-16x?

Answer 6

Next continue solving for x:

\(y^2 = -16x\)

\(-\dfrac{y^2}{16} = x\)

Answer 7

Isolating \(x\) is the correct thing to do here.

Answer 8

Sometimes \(x\) is expressed as

\(x = -\dfrac{1}{16}y^2\)

Answer 9

ok I got it thank you :)

Answer 10

You can use this formula as long as you have a focus and a directrix.

Answer 11

Just make sure you keep up with whether or not the directrix is equal to x or y.

Answer 12

In this case, if you notice, the directrix was x = 4. The directrix, in this case, can also be expressed as (4, y)

Answer 13

Do you know why that is?

Answer 14

Because it says it's equal to x?

Answer 15

I don't know exactly but it makes sense