# Thread: Paramatric equations and LOCUS!!!!

1. ## Paramatric equations and LOCUS!!!!

p is a variable point on the parabola x^2 = -4y. The tangent from P cuts the parabola x^2 = 4y at Q and R. Show that 3x^2 = 4y is the equation of the locus of the midpoint of the chord RQ.

I'm clueless.

2. Let P have co-ordinates $\displaystyle (p, -\frac{1}{4}p^2)$ on the lower parabola.

The slope of the tangent is $\displaystyle -\frac{1}{2}p$, and you can see from a sketch that this is equal to

$\displaystyle -\frac{1}{4}p^2$ divided by a crucial advance on the x-axis that turns out (as a result) to be

$\displaystyle \frac{1}{2}p$.

And from this you can deduce that the constant in the tangent-line equation is

$\displaystyle \frac{1}{4}p^2$.

So $\displaystyle y = -\frac{1}{2}p x + \frac{1}{4}p^2$

is an equation for the tangent line which you can set equal to the upper parabola that it has to (twice) intersect...

So solve that equation for x, to get the x-values (and after that the y's) of the points of intersection in terms of a, and then finally their mean average (mid-point) in terms of a. Hopefully that final y-value will be three-quarters the square of the x.

3. Thanks but OMG I have to look over that a couple of times to understand it.