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Math Help - Analytical Geometry

  1. #1
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    Analytical Geometry

    Given P(-2, 3) and circle (x-3)^2 + (y-3)^2 = 9, find the tangents (of the circle) through P.
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  2. #2
    MHF Contributor red_dog's Avatar
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    A line passing thruogh the point P and havinf the slope m has the equation

    y-3=m(x+2)

    The line must intersect the circle in a single point, so the system of equations formed by the equation of the line and the equation of the circle must have only one solution.

    \left\{\begin{array}{ll}(x-3)^2+(y-3)^2=9\\y-3=m(x+2)\end{array}\right.

    y=m(x+2)+3

    Replace y in the equation of the circle and you'll find a quadratic equation:

    (1+m^2)x^2+(4m^2-6)x+4m^2=0

    This equation must have equal roots. For this, the discriminant must be 0.

    \Delta=b^2-4ac=-64m^2+36=0\Rightarrow m=\pm\frac{3}{4}.

    Replace m in the equation of the line with these two numbers and you'll find two tangents line to the circle.
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