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Math Help - Proof Help

  1. #1
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    Proof Help

    Show that the equation of the normal to the parabola y^2=ax at the point (at^2, 2at0 is y+tx=2at+at^3.
    If this normal meets the curve again at the point (aT^2,2aT) show that
    t^2+tT+2=0
    and deduce that T^2 is greater than 8.

    My problem here is deducing the T^2 is greater than 8.
    Thanks
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  2. #2
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    Quote Originally Posted by arze View Post
    Show that the equation of the normal to the parabola y^2=ax at the point (at^2, 2at0 is y+tx=2at+at^3.
    If this normal meets the curve again at the point (aT^2,2aT) show that
    t^2+tT+2=0
    and deduce that T^2 is greater than 8.

    My problem here is deducing the T^2 is greater than 8.
    Thanks
    Its y^2=4ax right?

    Well since you've deduced t^2+Tt+2=0 it is easy to show that T^2 is greater than 8. Since the normal intersects the parabola in 2 points, the discriminant of the equation in t has to be greater than zero (for a tangent, discriminant = 0, discriminant is denoted by delta). Therefore, we get

    \Delta \textgreater0, <br />
 T^2 - 4(2) \textgreater0,<br />
T^2 \textgreater8
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