Find the center and radius of this equation: x^2 + 9x + y^2 – 5y – 8 = 0?

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  • x² + 9x + y² – 5y – 8 = 0
    x² + 9x + y² – 5y = 8
    x² + 9x + 20.25 + y² – 5y + 6.25 = 8 + 20.25 + 6.25
    (x + 4.5)² + (y – 2.5)² = 34.5

    Center (- 4.5, 2.5)
    ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

    Radius = 5.874
    ¯¯¯¯¯¯¯¯¯¯¯¯¯
     

  • The standard form of the equation of a circle is:
    (x – h)² + (y – k)² = r² where (h, k) is the center and r is the radius.
    The formula for h is the same as the axis of symmetry for a quadratic of the form x² + (b)x + c
    h = -b/2
    h = -9/2
    We can replace the x terms with (x – -9/2)² but, to keep the equation balanced, we must add h² to the right:
    (x – -9/2)² + y² – 5y -8 = 81/4
    Similarly, k is the axis of symmetry for a quadratic of the form y² + (b)y + c:
    k = -b/2
    k = 5/2
    Again, we can replace the y terms with (y – 5/2)² but we must add k² to the right:
    (x – -9/2)² + (y – 5/2)² -8 = 81/4 + 25/4
    Add 8 to both sides:
    (x – -9/2)² + (y – 5/2)² = 8 + 81/4 + 25/4

    (x – -9/2)² + (y – 5/2)² = 32/4 + 81/4 + 25/4

    (x – -9/2)² + (y – 5/2)² = {(√138)/2)}²

    The center is (-9/2, 5/2) and the radius is (√138)/2

  • First, make it into standard form which would be x(x+9) + y(y-5) = 8
    the standard form for a cicle is x(x-a) + y(y-b) = r^2
    (a,b) is the center and r is the radius
    Center: (-9,5)
    Radius: sqrt(8)= 2sqrt(2)

  • You just have to factor it and reduce it to the form (x-h)^2+(y-k)^2=r^2, you might have to add in some numbers on each side so that they become perfect squares and can be factorized into that form. The center is on the point (h,k) and the radius is r.

  • complete the square in x and y separately – start with
    x^2 + 9x + y^2 – 5y = 8
    x^2 + 9x + (9/2)^2 + y^2 – 5y + (-5/2)^2 = 8 + (9/2)^2 + (-5/2)^2
    the x and y parts can then be factored as –
    (x + 9/2)^2 + (y-5/2)^2 = 8 + (9/2)^2 + (-5/2)^2
    so the center is at (9/2 , -5/2) and the radius is
    r = sqrt( 8 + (9/2)^2 + (-5/2)^2 )

  • x^2 + 9x + y^2 – 5y – 8 = 0
    x^2 + y^2 + 9x – 5y – 8 = 0
    The center (h,k) and radius =r
    h = -a/2 = -9/2 = -4.5
    k= -b/2 = -(-5)/2 = 2.5
    c = h^2 + k^2 – r^2
    -8 = (-4.5^2) + (2.5^2) – r^2
    r = sqrt 34.5

  • (x² + 9x + 81/4) + (y² – 5y + 25/4) = 106/4

    (x + 9/2)² + (y – 5/2)² = [√ ( 53/2) ]²

    C (-9/2 , 5/2) , r = √ ( 53/2)

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