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Math Help - Parametric tangents and normals

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    Parametric tangents and normals

    P and Q are the points with parameters p and q on the parabola x=2at , y=at^2

    If pq = 1 and S is the focus of the parabola, show that

    1/SP + 1/SQ = 1/a

    Thanks
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  2. #2
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    Quote Originally Posted by deltaxray View Post
    P and Q are the points with parameters p and q on the parabola x=2at , y=at^2

    If pq = 1 and S is the focus of the parabola, show that

    1/SP + 1/SQ = 1/a
    The focus is at S = (0,a). If P = (2ap,ap^2) and Q = (2aq,aq^2) then SP^2 = 4a^2p^2 + a^2(1-p^2)^2 = a^2(1+p^2)^2 and so SP = a(1+p^2). Similarly SQ = a(1+q^2).

    Then \frac1{SP} + \frac1{SQ} = \frac1{a(1+p^2)} + \frac1{a(1+q^2)} = \frac{2+p^2+q^2}{a(1+p^2)(1+q^2)}. If pq=1 then you can write the numerator as 1+p^2+q^2+p^2q^2 = (1+p^2)(1+q^2), so the fraction simplifies to \frac1a.
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    Thank you
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