## lilsis76 2 years ago find the equation of the parabola with the vertex (0,0), and its focus the center of the circle with the equation: x^2 -8x +y^2 +15 = 0 fixed up: X^2 -8x = -Y^2 - 15/ completing the squar: -8/2 = -4^2 = 16 X^2 -8x +16 = -1(y - 15+16) (x-4)^2 = -1 (y-1) so that means the center is 4,1 right? but how can i figure the rest. this is all i got from my notes.

1. lilsis76

Now that I look at the problem, i dont even think I did it right

2. tkhunny

No. Nice try, through. This shows you are thinking about it and coming up with a plan. Seriously, good work. The circle is ONLY for finding the Focus of the parabola. You used it for a few other things. You correctly completed the square. This should have given the equation of a circle, $$(x-4)^{2} + y^{2} = 1$$, which is a circle of radius 1 with center at (4,0). This is ALL we need from the circle. We are done with it. We now have, for the desired parabola: Vertex: (0,0) Focus: (4,0) Can you find the equation of that opening-to-the-right parabola?

3. lilsis76

let me see if i can work it to a equation with....$(x-h)^{2} = 4p(y-k) \right?$

4. lilsis76

@tkhunny

5. tkhunny

Yes, but you already know h = k = 0. No need to mess with those.

6. lilsis76

okay um... hold on

7. lilsis76

ya, i got nothing :/ I keep getting the zero, like im answering it . i dont know how to find the equation

8. tkhunny

The distance from the vertex to the focus is 4. This makes the distance from the vertex to teh directrix also 4. You need to be able to find your '4p' from that information.

9. lilsis76

so if its 4, i divide by the 4p to get 1

10. tkhunny

Close. You're backwards. 'p' is the distance that we know. p = 4, then 4p = 16.

11. lilsis76

oh okay. so then the equation i put in the 16? this has been a long section and my brain is completelly fried badly

12. lilsis76

@tkhunny

13. tkhunny

Part of the task is knowing when you are done. Vertex: (0,0) Focus: (4,0) p (the mysterious parameter): 4 $$y^{2} = 16x$$ Done.

14. lilsis76

isnt that a different equation to another problem? it looks familar

15. lilsis76

how do i get the focus of 4,0?

16. tkhunny

That's where we started. Remember the cernter of the circle?

17. lilsis76

isnt that 0,0? right the center? cuz the vertex is.....is the point where the parabola starts

18. tkhunny

Youmust read the probelm statement a couple mroe time. You have become confused. Insta-Review ------ We now have, for the desired parabola: Vertex: (0,0) Focus: (4,0)

19. lilsis76

yes okay well i have this @tkhunny X^2 -8x = -Y^2 - 15/ completing the squar: -8/2 = -4^2 = 16 X^2 -8x +16 = -1(y - 15+16) (x-4)^2 = -1 (y-1) is the focus from the 16?

20. tkhunny

More review: The circle is ONLY for finding the Focus of the parabola. You used it for a few other things. You correctly completed the square. This should have given the equation of a circle, $$(x−4)^{2} +y^{2} =1$$ , which is a circle of radius 1 with center at (4,0). This is ALL we need from the circle. We are done with it.

21. lilsis76

okay, but how do we get the 16? i dont understand how that was found

22. tkhunny

More Review: The distance from the vertex to the focus is 4. Thus, p = 4 and 4p = 16

23. lilsis76

so that means.....you...divide by the 4 to get 4? an that is the focus?

24. tkhunny

Let's go sequentially, shall we. 1) We know we need a parabola. The posibilities are these: $$(x-h)^{2} = 4p(y-k)$$ or $$(y-k)^{2} = 4p(x-h)$$ 2) We know the vertex, (0,0). Now the possibilites are these: $$x^{2} = 4py$$ or $$y^{2} = 4px$$ -- Just substituting h = 0 and k = 0 and simplifying. 3) Using the circle hint, we determined the focus to be (4,0). Since this is just to the right of the vertex (0,0), we know the parabola opens to the right. Now the posibilities are these: $$y^{2} = 4px$$ -- No more "or". It's this kind and not the other. 4) Find 'p'. You need either the distance from the vertex to the focus or the vertex to the directrix. We have the former. p = 4 Thus: $$y^{2} = 4(4)x$$ -- Simply substituting the known value. 5) Simplify and we're done. $$y^{2} = 16x$$ -- Simply substituting the known value. One thing at a time. Slowly. Systematically. If you start getting confused, don't be afraid to start over and be more careful and more systematic.

25. lilsis76

i will thank you. i will go over these steps.