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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.

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  1. anonymous
    • one year ago
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    @Hero

  2. Hero
    • one year ago
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    @Nairz, what are your thoughts regarding this problem? Do you have an approach for solving this?

  3. anonymous
    • one year ago
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    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
    • one year ago
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    @Hero

  5. Hero
    • one year ago
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    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
    • one year ago
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    And that will give me the equation to the parabola?

  7. Hero
    • one year ago
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    Yes. Notice that in this case: Focus: \((x_1, y_1) = (0,-2)\) Directrix: \((x_2, y_2) = (x,2)\)

  8. Hero
    • one year ago
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    You'll have to do a bit of simplification after inserting the points in order to express the equation in standard form.

  9. anonymous
    • one year ago
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    yes. Thanks!

  10. Hero
    • one year ago
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    @Nairz, mind showing your work for this? Let's see what equation you come up with.

  11. anonymous
    • one year ago
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    ummm I got \[x=2\sqrt{2}\]

  12. anonymous
    • one year ago
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    ill show my work in a second

  13. anonymous
    • one year ago
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    \[\large (x-0)^2 + (y+2)^2 = (x-x)^2 + (y-2)^2\]

  14. anonymous
    • one year ago
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    \[\large x^2 + (y+2)^2 = (y+2)^2\]

  15. anonymous
    • one year ago
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    \[\large x^2 = (y-2)^2 - (y+2)^2\]

  16. anonymous
    • one year ago
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    \[\large x^2 = -8\]

  17. anonymous
    • one year ago
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    \[\large x=2\sqrt{2}\]

  18. Hero
    • one year ago
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    Okay, hang on a second.

  19. Hero
    • one year ago
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    \((y - 2)^2 - (y + 2)^2\) is actually "difference of squares". You should use the difference of squares formula to simplify that properly.

  20. anonymous
    • one year ago
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    i dont have a formula sheet in front of me

  21. Hero
    • one year ago
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    Difference of squares formula: \(a^2 - b^2 = (a + b)(a - b)\)

  22. Hero
    • one year ago
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    In this case a = y - 2 b = y + 2

  23. anonymous
    • one year ago
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    so its \[\large ((y-2)+ (y+2))((y-2)-(y+2))\]

  24. Hero
    • one year ago
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    Don't forget the \(x^2\) equals part

  25. anonymous
    • one year ago
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    simplified: \[\large x^2 = 2y(-4)\]

  26. anonymous
    • one year ago
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    so x^2 = -8y

  27. anonymous
    • one year ago
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    which isnt an option...

  28. Hero
    • one year ago
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    Finish isolating y

  29. anonymous
    • one year ago
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    y=x^2/-8

  30. Hero
    • one year ago
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    Note that the equation can also be written in the form \(y = -\dfrac{1}{8}x^2\)

  31. anonymous
    • one year ago
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    Ok... I just realized I was looking at the wrong problem's answers... That is there. Thanks

  32. anonymous
    • one year ago
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    Can you work through another one of these with me? Just to make sure I do it correctly?

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