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    The point P ( a*(t^2) , 2at ) lies on a parabola with equation y^2=4ax, where a is a positive constant.
    The equation of a normal to the parabola at P is y+xt = 2at+ (a* (t^3))

    The normal to the parabola at P meets the parabola again at Q

    Find. in terms of t, the co-ordinates of Q.




    can someone tell me what substitution has to be made to proceed with the problem and why we have to make the substitution? :confused:
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    your normal line at the point P is y+xt = 2at+ at^3 and this line meets the parabola at P and Q so both points lie on this line AND the parabola (aT^2,2aT). Here I've used T to show this T is variable, and the t in the first equation is fixed once we've picked a point P. Hence t \neq T.

    So on the parabola x=aT^2 and y=2aT.

    So make the substitutions in the equations above, into the first equation, giving (try it first, then look below:

    Spoiler:
    Show


    (2aT) + t(aT^2) = 2at + at^3 and solve for T in terms of T. Then, substituting the result into the equation of the parabola will give Q.17

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    (Original post by foolsihboy)
    your normal line at the point P is y+xt = 2at+ at^3 and this line meets the parabola at P and Q so both points lie on this line AND the parabola (aT^2,2aT). Here I've used T to show this T is variable, and the t in the first equation is fixed once we've picked a point P. Hence t \neq T.

    So on the parabola x=aT^2 and y=2aT.

    So make the substitutions in the equations above, into the first equation, giving (try it first, then look below:

    Spoiler:
    Show


    (2aT) + t(aT^2) = 2at + at^3 and solve for T in terms of T. Then, substituting the result into the equation of the parabola will give Q.17

    Im finding it a little difficult to understand why one t is variable and one is not in the same equation...

    Isn't the equation for the normal to a parabola, y+xt = 2at+ at^3 , for all the co-ordinates of the normal where each of these co-ordinates are generated from a distinct t value?
    thanks!
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    (Original post by confuzzled92)
    Im finding it a little difficult to understand why one t is variable and one is not in the same equation...

    Isn't the equation for the normal to a parabola, y+xt = 2at+ at^3 , for all the co-ordinates of the normal where each of these co-ordinates are generated from a distinct t value?
    thanks!
    Do You think that P lies on the parabola for all t?
 
 
 
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