1. ## Proof Help

Show that the equation of the normal to the parabola $\displaystyle y^2=ax$ at the point $\displaystyle (at^2, 2at0$ is $\displaystyle y+tx=2at+at^3$.
If this normal meets the curve again at the point $\displaystyle (aT^2,2aT)$ show that
$\displaystyle t^2+tT+2=0$
and deduce that $\displaystyle T^2$ is greater than 8.

My problem here is deducing the $\displaystyle T^2$ is greater than 8.
Thanks

2. Originally Posted by arze
Show that the equation of the normal to the parabola $\displaystyle y^2=ax$ at the point $\displaystyle (at^2, 2at0$ is $\displaystyle y+tx=2at+at^3$.
If this normal meets the curve again at the point $\displaystyle (aT^2,2aT)$ show that
$\displaystyle t^2+tT+2=0$
and deduce that $\displaystyle T^2$ is greater than 8.

My problem here is deducing the $\displaystyle T^2$ is greater than 8.
Thanks
Its $\displaystyle y^2=4ax$ right?

Well since you've deduced $\displaystyle t^2+Tt+2=0$ it is easy to show that $\displaystyle T^2$ is greater than 8. Since the normal intersects the parabola in 2 points, the discriminant of the equation in t has to be greater than zero (for a tangent, discriminant = 0, discriminant is denoted by delta). Therefore, we get

$\displaystyle \Delta \textgreater0, T^2 - 4(2) \textgreater0, T^2 \textgreater8$