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Math Help - How can I determine this is a parabola? (by finding its standard form)

  1. #1
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    How can I determine this is a parabola? (by finding its standard form)

    The equation is:

    3x^2-20y^2=80y+48

    I know it is a hyperbola once I check with Wolfram Alpha. But, I need a definite way of knowing it is.

    I tried setting it up in the standard form by trying to isolate 48.

    3x^2-20y^2-80y=48

    Then divide by 48 to make the right side equal 1 like the standard form for it is.

    3x^2/48-(20y^2-80y)/48=1

    But that is still not its standard form because I don't know what y^2 is... how can I get it to be just y^2?
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  2. #2
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    Quote Originally Posted by thyrgle View Post
    The equation is:

    3x^2-20y^2=80y+48

    I know it is a hyperbola once I check with Wolfram Alpha. But, I need a definite way of knowing it is.

    I tried setting it up in the standard form by trying to isolate 48.

    3x^2-20y^2-80y=48

    Then divide by 48 to make the right side equal 1 like the standard form for it is.

    3x^2/48-(20y^2-80y)/48=1

    But that is still not its standard form because I don't know what y^2 is... how can I get it to be just y^2?


    You first need to complete the square in y: 20y^2+80y=20(y^2+4y)=20(y+2)^2-80 , and then:

    \displaystyle{3x^2-20y^2=80y+48\Longrightarrow 3x^2-20(y+2)^2+80=48\Longrightarrow 3x^2-20(y+2)^2=-32\Longrightarrow

     \frac{(y+2)^2}{8/5}-\frac{x^2}{32/3}=1

    Tonio
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  3. #3
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    Quote Originally Posted by thyrgle View Post
    3x^2-20y^2=80y+48
    Re-arranging:
    20y^2 + 80y - 3x^2 + 48 = 0
    Clearly of form Ax^2 + Bx + C = 0
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  4. #4
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    "How can I determine this is a parabola?"

    first note that tonio shows how to derive 'standard' form. Wilmer had shown'general form'.

    full conic general form is Ax^2+Bxy+Cy^2+Dx+Ey+F

    a quick way to determine if you have parabola,hyperbola or ellipse is to look at A and C terms.

    if there is only one second power term you have a parabola.
    if C is negative you have a hyperbola.
    if A and C are positive you have ellipse.

    in standard form just look at the sign in between the terms '-'=hyperbola '+'=ellipse

    to determine algebraically with NO B term.
    A=C circle
    AC > 0 = ellipse
    AC = 0 = parabola
    AC < 0 = hyperbola

    to determine algebraically with B term use the discriminate.
    B^2-4AC
    discriminate < 0 = ellipse
    discriminate = 0 = parabola
    discriminate > 0 = hyperbola
    Last edited by skoker; June 7th 2011 at 07:34 PM.
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  5. #5
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    Quote Originally Posted by Wilmer View Post
    Re-arranging:
    20y^2 + 80y - 3x^2 + 48 = 0
    Clearly of form Ax^2 + Bx + C = 0
    What????
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  6. #6
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    Charlie Brown, Charlie Brown
    He's a clown, that Charlie Brown
    He's gonna get caught
    Just you wait and see
    Why's HallsofIvy always pickin' on me !
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  7. #7
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    Hello, thyrgle

    Is this a parabola? . 3x^2-20y^2\:=\:80y+48
    Answer: no.


    Skoker explained the equation of the general conic equation,
    . . which includes conics which have been "rotated".

    If there is no xy-term, the conic is "horizontal" or "vertical".
    . . Its axis is parallel to a coordinate axis.

    Then the equation can be "eyeballed" to identify the conic.


    The general equation of this conic is: . Ax^2 + By^2 + Cx + Dy + E \:=\:0

    We are concerned with A and B only, the coefficients of x^2 and y^2.


    If A = 0 or B = 0 (but not both), we have a parabola

    If A = B, we have a circle.

    If A \ne B and A,B have the same sign: ellipse.

    If A,B have opposite signs: hyperbola.


    Note: the above forms include degenerate and imaginary conics.

    For example: . x^2 + y^2 \:=\:0 is a degenerate circle.

    And: . 4x^2 + y^2 + 9\:=\:0 is an imaginary ellipse.

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