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Math Help - Hyperbola Equation

  1. #1
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    Hyperbola Equation

    Write the equation of the hyperbola with foci at (5,0)
    and (-5,0) and eccentricity 5/3

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  2. #2
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    Hyperbola Equation...Part 2

    Find an equation for the hyperbola with vertices
    at (2, 15) and (2, 1), and having eccentricity e = 17/8
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  3. #3
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    You could use the general form  y= \frac{a}{x-b}+c and substitute the information you have.
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  4. #4
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    Quote Originally Posted by pickslides View Post
    You could use the general form  y= \frac{a}{x-b}+c and substitute the information you have.
    Oops, that's not the right type of hyperbola. The form we need is
    \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1
    or
    \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1.

    In our case, for the 1st question,
    Write the equation of the hyperbola with foci at (5,0)
    and (-5,0) and eccentricity 5/3
    Since the two foci lie on a horizontal line, we use this form of the hyperbola:
    \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1.

    It's also easy to see that the center = (h , k) = (0, 0).

    Remember that the distance from the center to one of the foci is c, and
    remember that eccentricity is e = \frac{c}{a}.

    Also, use the Pythagorean relation c^2 = a^2 + b^2 to find b. I'll let you go on from here.

    Find an equation for the hyperbola with vertices
    at (–2, 15) and (–2, –1), and having eccentricity e = 17/8
    The two vertices lie on a vertical line, so the form of the hyperbola to use is
    \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1.

    The midpoint between the two vertices is the center (h, k). The distance between the center and one of the vertices is a. Use the eccentricity formula and Pythagorean relation above to find c and b.


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  5. #5
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    Hi Magentarita,

    The general equation of a hyperbola is:

    \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

    For this formulation the directrix is x = a/e and foci at (ae,0). Also b^2 = a^2(e^2 - 1). With this information

    5 = a.  \frac{5}{3}

    giving a = 3

    Also 3 = b^2(\frac{25}{9} - 1)

    giving b^2 = \frac{27}{16}

    Therefore the equation is:

    \frac{x^2}{9} - \frac{16.y^2}{27} = 1

    3x^2 - 16y^2 = 27

    The thing is there are various formulations of the equation for a hyperbola and with the right choice of the constants, the question almost answers itself.
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