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Need help for a parametrics question please (1 Viewer)

APOLLO2609

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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!!
 

pikachu975

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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)
 

APOLLO2609

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

pikachu975

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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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