math

Let A = (−7,−4) and B = (7,4), and consider the equation PA•PB = 0. Describe
the configuration of all points P = (x, y) that solve this equation.

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  1. Assuming that PA and PB are vectors and PA•PB = 0
    is the dot products,
    vector PA = (-7-x,-4-y) and PB = (7-x,4-y)
    then PA•PB = 0 ---> (-7-x,-4-y)•(7-x,4-y) = 0
    -49 + 7x - 7x + x^2 - 16 + 4y - 4y + y^2 = 0
    x^2 + y^2 = 65

    Which is the equation of a circle, with centre at (0,0) and radius √65
    This makes perfect sense, since A and B are the end points of a
    diameter. Any point P on a circle joined to the endpoints of a diameter of that circle
    creates a right-angled triangle.

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