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Thread: Need help for a parametrics question please

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    Need help for a parametrics question please

    The question is:

    T(2at, at2) is a point on the parabola x2=4ay with focus F(0,a). The line with gradient t through F meets the parabola at the points P and Q. M is the midpoint of PQ. Show that M has coordinates (2at, 2at2 + a).

    Thank you!!

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    Re: Need help for a parametrics question please

    Quote Originally Posted by APOLLO2609 View Post
    The question is:

    T(2at, at2) is a point on the parabola x2=4ay with focus F(0,a). The line with gradient t through F meets the parabola at the points P and Q. M is the midpoint of PQ. Show that M has coordinates (2at, 2at2 + a).

    Thank you!!
    y - a = tx
    y = tx+a

    Sub this into parabola equation to find P and Q:
    x^2 = 4a(tx+a)
    x^2 - 4atx - 4a^2 = 0

    Midpoint of P and Q is M which has x value of x = (x1+x2)/2 where x1 and x2 are x-coordinates of P and Q, which is sum of roots of above equation divided by 2:

    x = 4at/2 = 2at

    Sub this into the line equation:

    y = t(2at)+a
    y = 2at^2 + a

    Hence M(2at, 2at^2 + a)
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    Re: Need help for a parametrics question please

    Thank you, I appreciate the help. Simple question, I just didn't recognise x1 and x2 to be the roots haha. Thanks again!
    pikachu975 likes this.

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    Re: Need help for a parametrics question please

    Quote Originally Posted by APOLLO2609 View Post
    Thank you, I appreciate the help. Simple question, I just didn't recognise x1 and x2 to be the roots haha. Thanks again!
    No worries. I learnt this trick from my teacher when doing a past paper. It's really useful so keep it in mind - I used it in 2017 hsc
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