Another question about the parabola...

**free_to_fly** · December 16th 2007, 09:04 AM

Lol I seem to be stuck with a lot of questions to do with the parabola.

Q: Show that the normals at P (at^2, 2at) and Q (as^2,2as)on the parabola with equation y^2=4ax meet at the point R where the coordinates of R are
[a(t^2+ts+s^2+2)], -ats(t+s)]. This part of the question I've managed to do. It's the next part that I'm stuck on: Given that the line PQ passes through the focus S(a, 0), show that as t and s vary, R lies on the curve with equation y^2=a(x-3a).

I tried to eliminate the variables t and s, but was unsuccessful. Can someone help please?

Btw, can someone let me know what topic this post should come under for future reference?

**red_dog** · December 17th 2007, 08:35 AM

The line PQ has the equation
$\displaystyle\frac{x-at^2}{a(s^2-t^2}=\frac{y-2at}{2a(s-t)}$
or $\displaystyle\frac{x-at^2}{s+t}=\frac{y-2at}{2}$ .
If the line passes through the focus, then we have
$\displaystyle\frac{a-at^2}{s+t}=-\frac{2at}{2}\Rightarrow st=-1$ .
The coordinates of R are
$x=a(t^2+st+s^2+2)\Rightarrow x=a(t^2+s^2+1)$
$y=-ats(t+s)\Rightarrow y=a(t+s)$ .
Now, we have
$y^2=a^2(t^2+s^2+2st)=a^2(t^2+s^2+1-3)=a^2\left(\frac{x}{a}-3\right)=a(x-3a)$

Thread: Another question about the parabola...

LinkBack

Thread Tools

Display

Another question about the parabola...

Similar Math Help Forum Discussions

Parabola question

Parabola Question

parabola question

Parabola(?) Question

A Question About a Parabola

Search Tags