Post a New Question

Algebra II

posted by on .

Howndar do I write x^2-4y^2-4x-8y=36 in standard form?

  • Algebra II - ,

    Hyperbola (x-2)^2-4(y-1)^2=36

    I have that one on my exam too.

  • Algebra II - ,

    The General Equation for a Conic Sections:

    A x ^ 2 + B x y + C y ^ 2 + D x + E y + F = 0

    In this case:

    x ^ 2 - 4 y ^ 2 - 4 x - 8 y = 36 Subtract 36 to both sides

    x ^ 2 - 4 y ^ 2 - 4 x - 8 y - 36 = 0


    1 * x ^ 2 + 0 * B x y - 4 * y ^ 2 - 4 * x - 8 * y - 36 = 0


    A = 1

    B = 0

    C = - 4

    D = - 4

    E = - 8

    F = - 36


    The discriminant B ^ 2 - 4 A C will identify which conic section it is.


    If the discriminant is positive, the section is a hyperbola.

    If it is negative, the section is an ellipse.

    If it is zero, the section is a parabola.


    B ^ 2 - 4 A C = 0 ^ 2 - 4 * 1 * ( - 4 ) = 0 + 16 = 16

    The discriminant is positive.

    Your line is a hyperbola.


    Equation of hyperbola in standard form :

    ( x - h ) ^ 2 / a ^ 2 - ( y - k ) ^ 2 / b ^ 2 = 1


    x ^ 2 - 4 y ^ 2 - 4 x - 8 y = 36

    x ^ 2 - 4 x - 4 y ^ 2 - 8 y = 36

    ( x ^ 2 - 4 x ) - 4 ( y ^ 2 + 2 y ) = 36


    The process involves completing the square separately for the x and y variables.


    ____________________________________


    ( x - 2 ) ^ 2 = x ^ 2 - 2 * x * 2 + 2 ^ 2

    ( x - 2 ) ^ 2 = x ^ 2 - 4 x + 4 Subtract 4 to both sides

    ( x - 2 ) ^ 2 - 4 = x ^ 2 - 4 x + 4 - 4

    ( x - 2 ) ^ 2 - 4 = x ^ 2 - 4 x

    x ^ 2 - 4 x = ( x - 2 ) ^ 2 - 4



    ( y + 1 ) ^ 2 = y ^ 2 + 2 * y * 1 + 1 ^ 2

    ( y + 1 ) ^ 2 = y ^ 2 + 2 y + 1 Subtract 1 to both sides

    ( y + 1 ) ^ 2 - 1 = y ^ 2 + 2 y + 1 - 1

    ( y + 1 ) ^ 2 - 1 = y ^ 2 + 2 y

    y ^ 2 + 2 y = ( y + 1 ) ^ 2 - 1

    - 4 * ( y ^ 2 + 2 y ) = - 4 * [ ( y + 1 ) ^ 2 - 1 ]

    - 4 * ( y ^ 2 + 2 y ) = - 4 ( y + 1 ) ^ 2 + 4


    __________________________


    x ^ 2 - 4 x - 4 y ^ 2 - 8 y = 36

    x ^ 2 - 4 x - 4 ( y ^ 2 + 2 y ) = 36

    ( x ^ 2 - 4 x ) - 4 ( y ^ 2 + 2 y ) = 36

    ( x - 2 ) ^ 2 - 4 - 4 ( y + 1 ) ^ 2 + 4 = 36

    ( x - 2 ) ^ 2 - 4 ( y + 1 ) ^ 2 = 36 Divide both sides by 36

    ( x - 2 ) ^ 2 / 36 - 4 ( y + 1 ) ^ 2 / 36 = 1

    ( x - 2 ) ^ 2 / 36 - 4 ( y + 1 ) ^ 2 / ( 4 * 9 ) = 1


    ( x - 2 ) ^ 2 / 36 - ( y + 1 ) ^ 2 / 9 = 1

Answer This Question

First Name:
School Subject:
Answer:

Related Questions

More Related Questions

Post a New Question