# Math Help - Co-ordinate Geometry Parabola equation

1. ## Co-ordinate Geometry Parabola equation

A line joining the points A(-4a,0) to the point P(a $t^2$,2at), where a is a positive constant.
As t varies the locus of the midpoint of the line AP is a parabola C.

a) Find the cartesian equation of C

i dont quite know how to work this out and what deos it mean when the question says "As t varies"?

2. Originally Posted by Stylis10
A line joining the points A(-4a,0) to the point P(a $t^2$,2at), where a is a positive constant.
As t varies the locus of the midpoint of the line AP is a parabola C.

a) Find the cartesian equation of C

i dont quite know how to work this out and what deos it mean when the question says "As t varies"?
1. Calculate the coordinates of the midpoint $M\left(\dfrac{at^2-4a}2\ ,\ \dfrac{0+2at}2\right)$

2. You now have a curve in parametric form:

$c:\left\{\begin{array}{l}x=\dfrac{at^2-4a}2 \\ y = at\end{array}\right.$ ........ The variable for both equations is t (that means t varies!)

3. Solve the 2nd equation for t and substitute t in the first equation by this term:

$x=\dfrac{a\left( \frac ya \right)^2-4a}2$ ........ Rearrange this equation:

$x=\dfrac{y^2-4a^2}{2a}~\implies~\boxed{y^2=2ax+4a^2}$

Since a > 0 this is the equation of a parabola opening to the right.