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Advanced Math

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A circle of center (-3 , -2) passes through the points (0 , -6) and (a , 0). Find a.

The triangle bounded by the lines y = 0, y = 2x and y = -0.5x + k, with k positive, is equal to 80 square units. Find k.


When the polynomial P(x) = x3 + 3x2 -2Ax + 3, where A is a constant, is divided by x2 + 1 we get a remainder equal to -5x. Find A.


And would like to see work that can justify the answer. Thank you!

  • Advanced Math - ,

    Hints:
    (a)
    find the radius r which corresponds to the distance between (-3,-2) and (0,-6). Then
    (-3-a)^2+(-2-0)^2 = r^2

    (b)
    y=0 is horizontal (1)
    y=2x (2) is a line through the origin sloping upwards
    y=-0.5x+k (3) is a line sloping downwards with x-intercept: -0.5x+k=0, or x=2k.
    You'll need to find the intercept between lines (2) and (3) and determine the height of the triangle.

    (3)
    This means that (x^2+1) divides P(x)-5x, or
    P(x)-5x = (x^2+1)(x+3)=
    =x^3+3*x^2+x+3
    Therefore
    P(x)=x^3+3*x^2+x+3 + 5x
    =x^3+3*x^2+6x+3
    => A=-3

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