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Math Help - Tangent to circle

  1. #1
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    Tangent to circle

    The line pair joining the origin to the points A and B of intersection of the conic ax^2+by^2=1 and lx+my=1 is
    (a-l^2)x^2-2lmxy+(b-m^2)y^2=0
    If angle AOB is a right angle, show that AB touches the circle
    (a+b)(x^2+y^2)=1
    I have no idea how to do this. Any pointers?
    Thanks a million!
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  2. #2
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    If angle AOB is a right angle,

    (a-l^2) + (b-m^2) = 0

    Or a+b = l^2 + m^2

    If lx + my = 1 is the tangent to a circle, the distance of this line from the center is equal to the radius.

    \frac{1}{l^2 + m^2} = r^2

    \frac{1}{a+b} = r^2

    So the equation of the circle is

    x^2 + y^2 = \frac{1}{a+b}
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