Find the standard form of the equation of the parabola with a focus at (0, -2) and a directrix at y = 2.

Question

anonymous · Answer

@Hero

hero · Answer

@Nairz, what are your thoughts regarding this problem? Do you have an approach for solving this?

anonymous · Answer

Im assuming the vertex is the origin since the directrix and the focus are both 2 units away from the origin along the same line.

anonymous · Answer

@Hero

hero · Answer

Consider this: If you have two points, the focus $(x_1, y_1)$ and the directrix $(x_2, y_2)$ then you can insert those points in to the following formula to find the standard form of the equation of a parabola:

$(x - x_1)^2 + (y - y_1)^2 = (x - x_2)^2 + (y - y_2)^2$

anonymous · Answer

And that will give me the equation to the parabola?

hero · Answer

Yes. Notice that in this case:

Focus: $(x_1, y_1) = (0,-2)$
Directrix: $(x_2, y_2) = (x,2)$

hero · Answer

You'll have to do a bit of simplification after inserting the points in order to express the equation in standard form.

anonymous · Answer

yes. Thanks!

hero · Answer

@Nairz, mind showing your work for this? Let's see what equation you come up with.

anonymous · Answer

ummm I got $$x=2\sqrt{2}$$

anonymous · Answer

ill show my work in a second

anonymous · Answer

$$\large (x-0)^2 + (y+2)^2 = (x-x)^2 + (y-2)^2$$

anonymous · Answer

$$\large x^2 + (y+2)^2 = (y+2)^2$$

anonymous · Answer

$$\large x^2 = (y-2)^2 - (y+2)^2$$

anonymous · Answer

$$\large x^2 = -8$$

anonymous · Answer

$$\large x=2\sqrt{2}$$

hero · Answer

Okay, hang on a second.

hero · Answer

$(y - 2)^2 - (y + 2)^2$ is actually "difference of squares".  You should use the difference of squares formula to simplify that properly.

anonymous · Answer

i dont have a formula sheet in front of me

hero · Answer

Difference of squares formula:
$a^2 - b^2 = (a + b)(a - b)$

hero · Answer

In this case 
a = y - 2
b = y + 2

anonymous · Answer

so its $$\large ((y-2)+ (y+2))((y-2)-(y+2))$$

hero · Answer

Don't forget the $x^2$ equals part

anonymous · Answer

simplified: $$\large x^2 = 2y(-4)$$

anonymous · Answer

so x^2 = -8y

anonymous · Answer

which isnt an option...

hero · Answer

Finish isolating y

anonymous · Answer

y=x^2/-8

hero · Answer

Note that the equation can also be written in the form
$y = -\dfrac{1}{8}x^2$

anonymous · Answer

Ok... I just realized I was looking at the wrong problem's answers... That is there. Thanks

anonymous · Answer

Can you work through another one of these with me? Just to make sure I do it correctly?