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Math Help - Prove tangent bisects line

  1. #1
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    Prove tangent bisects line

    P is a point on an ellipse whose major axis is AB the tangent at P meets the minor axis at Q. PA and PB cut the minor axis at R and T. Prove that Q bisects RT.

    I put P(a\cos\theta,b\sin\theta) A=(a,0) and B=(-a,0)
    equation of tangent at P
    bx\cos\theta+ay\sin\theta=ab
    Point where tangent cuts minor axis, when x=o
    y=\frac{b}{\sin\theta}{
    Q(0,\frac{b}{\sin\theta})

    equation AP
    y-0=\frac{b\sin\theta}{a\cos\theta-a}(x-a)
    when x=0 y=-\frac{b\sin\theta}{\cos\theta-1}
    R(0,-\frac{b\sin\theta}{\cos\theta-1})

    equation BP
    y-0=\frac{b\sin\theta}{a\cos\theta+a}(x+a)
    when x=0, y=\frac{b\sin\theta}{\cos\theta+1}
    T(0,\frac{b\sin\theta}{\cos\theta+1})

    RT=-\frac{b\sin\theta}{\cos\theta-1}-\frac{b\sin\theta}{\cos\theta+1}
    =-\frac{2b\sin\theta\cos\theta}{\cos^2\theta-1}

    RQ=-\frac{b\sin\theta}{\cos\theta-1}-\frac{b}{\sin\theta}
    =-\frac{b(2+\cos\theta)\sin\theta}{\cos^2\theta-1}
    Now I don't know what to do. I can't show that RT=2RQ
    Thanks!
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  2. #2
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    You have found the co-ordinates of R and T.
    Find the mid point R and T. That will be the co-ordinates of Q.
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