Locus Problems (1 Viewer)

yeyjasminetime · Jul 28, 2013

Hi guys does anyone know how to do this?
THANKS!

The tangent at P(2at,at²) on the parabola x²=4ay meets the x-axis at T

Show that as t varies, the locus of M is the parabola 2x²=9xy

Kurosaki · Jul 28, 2013

yeyjasminetime said:
Hi guys does anyone know how to do this?
THANKS!

The tangent at P(2at,at²) on the parabola x²=4ay meets the x-axis at T

Show that as t varies, the locus of M is the parabola 2x²=9xy

What is M? the midpoint of PT?

yeyjasminetime · Jul 28, 2013

Oh oops!
Yes that's right M is the midpoint of PT

Kurosaki · Jul 28, 2013

Hmm, should that be 9ay?

yeyjasminetime · Jul 28, 2013

Oh geez I'm so sorry I should edit my threads before i post them
omg

Yes that is meant to be 4ay

yeyjasminetime · Jul 28, 2013

Sorry! If i've wasted your time

Kurosaki · Jul 28, 2013

Differentiating the parabola gives y'=x/2a. fiddle around with that you get to the tangent equation being y=tx-at^2
You then find that T is (at,0).
Therefore $M(\frac{3at}{2},\frac{at^2}{2})$
So therefore, $x=\frac{3at}{2}$ . $y=\frac{at^2}{2}$
Rearrange x to get $\frac{2}{at}=\frac{3}{x}$
$y=\frac{at^2}{2}=\frac{\frac{t}{2}}{at}$
Then $y=\frac{tx}{3}$
$y=\frac{\frac{2x^2}{3a}}{3}=\frac{2x^2}{9a}$

yeyjasminetime · Jul 28, 2013

Thank you so much, Kurosaki!

I just don't understand how that shows t varies.

If you wouldn't mind could you explain the thought process from this part

"Rearrange x to get ..."

Locus Problems (1 Viewer)

yeyjasminetime

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Kurosaki

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yeyjasminetime

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