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

1. anonymous

@Hero

2. Hero

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

3. anonymous

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.

4. anonymous

@Hero

5. Hero

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

6. anonymous

And that will give me the equation to the parabola?

7. Hero

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

8. Hero

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

9. anonymous

yes. Thanks!

10. Hero

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

11. anonymous

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

12. anonymous

ill show my work in a second

13. anonymous

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

14. anonymous

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

15. anonymous

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

16. anonymous

$\large x^2 = -8$

17. anonymous

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

18. Hero

Okay, hang on a second.

19. Hero

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

20. anonymous

i dont have a formula sheet in front of me

21. Hero

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

22. Hero

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

23. anonymous

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

24. Hero

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

25. anonymous

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

26. anonymous

so x^2 = -8y

27. anonymous

which isnt an option...

28. Hero

Finish isolating y

29. anonymous

y=x^2/-8

30. Hero

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

31. anonymous

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

32. anonymous

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