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A point p(x,y) is on the line through A(-4,4) and B(5,2).Find (a) the coordinates of P given that the segment AB is extended through B to P so that P is twice as far from A as far from B,and (b) the coordinates of P given that AB is extended through A to P so that P is three times as far from B as from A.

  • geometry -

    a. A(-4,4), (B(5,2), P(x,y).

    AP = 2BP,
    x-(-4) = 2(x-5),
    x + 4 = 2x -10,
    2x -x = 4 + 10 = 14,
    X = 14.

    y -4 = 2(y-2),
    y -4 = 2y -4,
    2y - y = -4 + 4 = 0,
    Y = 0.

    Solution set: P(14,0).

    b. P(x,y), A(-4,4), B(5,2).

    PB = 3PA,
    5 - x = 3(-4 -x),
    5 - x = -12 -3x,
    -x + 3x = - 12 - 5,
    2x = - 17,
    X = -17/2 = -8 1/2.

    2 - y = 3(4-y),
    2 - y = 12 - 3y,
    - y + 3y = 12 - 2,
    2y = - 10,
    Y = 5

    Solution set: P(-8 1/2,5).

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