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Math Help - Writing equation for hyperbola

  1. #1
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    Writing equation for hyperbola

    Write the equation of a hyperbola satisfying the given condition:
    with asymptotes y=-x and y=x+2 and vertical tranverse axis length of 4.

    vertical tranverse axis

    so hyperbola opens up and down

    length of 4

    so 2b=4, b=2

    intersection of y=-x and y=x+2 is (-1,1)

    so centre is (-1,1)

    The equation so far is:

    (x+1) / a - (y-1) /2 = -1

    But what is the value of a? How do you find a?

    Thanks.
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by shenton View Post
    Write the equation of a hyperbola satisfying the given condition:
    with asymptotes y=-x and y=x+2 and vertical tranverse axis length of 4.

    vertical tranverse axis

    so hyperbola opens up and down

    length of 4

    so 2b=4, b=2

    intersection of y=-x and y=x+2 is (-1,1)

    so centre is (-1,1)

    The equation so far is:

    (x+1) / a - (y-1) /2 = -1

    But what is the value of a? How do you find a?

    Thanks.
    One way:
    Solve for y:
    y = (1/a)*[a (+/-) 2*sqrt{(x + 1)^2 + a^2}]

    For large x this turns into:
    y = (1/a)*[a (+/-) 2x] = (+/-)(x/a) + 1 = (+/-)(2x/a)

    The asymptotes are (for large x) y = (+/-)x

    Thus a = 2.

    Below is the graph of (x + 1)^2/4 - (y - 1)^2/4 = -1

    -Dan

    PS In case anyone saw this whilst I edited it a zillion times, this is my final answer! (And I'm stickin' to it!)
    Attached Thumbnails Attached Thumbnails Writing equation for hyperbola-hyperbola.jpg  
    Last edited by topsquark; March 18th 2007 at 11:09 AM.
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  3. #3
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    Hello, shenton!

    Write the equation of a hyperbola satisfying the given condition:
    with asymptotes y = -x and y = x + 2 and vertical tranverse axis length of 4.

    Vertical tranverse axis, so hyperbola opens up and down.

    Length of 4: so 2b = 4, b = 2

    Intersection of y = -x and y = x + 2 is (-1,1) . . . so centre is (-1,1) . Good!

    The equation so far is:
    . . (x + 1)/a - (y - 1)/2 .= .- 1 . Yes!

    But what is the value of a?

    Recall that the slopes of the asymptotes are: .b/a

    The given asymptotes have slopes 1
    . . Hence: .b .= .a

    Therefore: .a = 2

    The equation is: .(y - 1)/4 - (x + 1)/4 .= .1

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