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Math Help - I am having trouble figuring out how to solve a conic line collision problem.

  1. #1
    Junior Member
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    I am having trouble figuring out how to solve a conic line collision problem.

    The problem in my book about conics states:

    "A hyperbolic mirror is a mirror in the shape of one branch of a hyperbola. Such a mirror reflects light rays directed at one focus toward the other focus. Suppose a hyperbolic mirror is modeled by the upper branch of the hyperbola with equation: y^2/9-x^2/16=1. A light source is located at (-10, 0). Where should the light from the source hit the mirror so that the light will be reflected to (0, -5)?"

    I am confused an do not know where to start. Sorry for such an ambiguous question, but basically, I just need a subtle push to the right start.
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  2. #2
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    Let P(x1, y1) be the point of incidence of light. Let OP be the incident ray and PI be the reflected ray.
    Find the slope of the tangent to the hyperbola at P. And from that find the slope of the normal at P.
    The normal is the angle bisector of OPI.
    Find the equation of OP and PI. From that find the equation of the angle bisector of angle OPI. Find the slope of the angle bisector.
    Equate it with that of the normal. You will get a relation between x1 and y1. Substitute this in the equation of the hyperbola to find the co-ordinates of P.
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