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Focus of a parabola

  1. Oct 27, 2009 #1

    Mentallic

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    1. The problem statement, all variables and given/known data
    This problem is from the Australian HSC mathematics extension 2 exam. Q6ciii)

    It states:

    Find the focus, S, of the parabola [tex]y^2=r^2+c^2-2cx[/tex] where r and c are constants.


    3. The attempt at a solution
    I couldn't figure out how to convert this into the parabola focus form (which, from the top of my head) might be [tex](x-x_o)^2=4a(y-y_o)[/tex] for the focus [tex]S(x_o,y_o+a)[/tex]

    How is this done?
     
  2. jcsd
  3. Oct 27, 2009 #2

    lanedance

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    notice the form of x & y are reversed, so you will be looking the equation in the form

    [tex](y-y_o)^2=4a(x-x_o)[/tex]

    with cordinates in the focus changed as well

    note that the co-efficient of y is zero, which implies y_o must be zero in the above... see how you go form here
     
  4. Oct 27, 2009 #3

    Mark44

    Staff: Mentor

    In your formula, the parabola's vertex is at (x0, y0) and it opens upward if a > 0. The parabola you're working with opens to the left if c > 0, and to the right if c < 0.

    Can you put your parabola in the form (y - y)2 = 4a(x - x0)?
     
  5. Oct 27, 2009 #4

    Mentallic

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    Oh ok so [tex]r^2+c^2-2cx\equiv 4a(x-x_o)[/tex]

    [tex]LHS=-2c(x-b)=-2cx+2cb[/tex]

    therefore [tex]2cb=r^2+c^2[/tex]

    then [tex]b=\frac{r^2+c^2}{2c}[/tex]

    Finally, [tex]y^2=-2c(x-\frac{r^2+c^2}{2c})[/tex]

    So then the focus is [tex]S(\frac{r^2+c^2}{2c}-\frac{c}{2},0)[/tex]

    Is this correct?

    edit: simplified, [tex]S(\frac{r^2}{2c},0)[/tex]
     
  6. Oct 27, 2009 #5

    lanedance

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    i haven't checked the original focus defintion, but fr0m what you give I get

    [tex]y^2=r^2+c^2-2cx[/tex]

    [tex](y - 0)^2=4 \frac{1}{4} (r^2+c^2-2cx)[/tex]

    [tex](y - 0)^2=4 (\frac{-c}{2})(x-\frac{r^2+c^2}{2c})[/tex]

    so
    [tex]a = \frac{-c}{2}[/tex]

    [tex]x_0 = \frac{r^2+c^2}{2c} [/tex]

    [tex]y_0 = 0 [/tex]

    then
    [tex] focus = ((x_0 + a), y_)) [/tex]

    [tex] focus = ((\frac{r^2+c^2}{2c} + \frac{-c}{2}), 0) [/tex]

    which look the same
     
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