Need help for a parametrics question please (1 Viewer)

[APOLLO2609]

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]

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]

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

 

[pikachu975]

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