Browse Questions

# Find the relation between $'t_1'$ meets the parabola again at $'t_2'$ for the parabola $y^2=4ax$?

$\begin{array}{1 1}(a)\;t_1=-t_2-\large\frac{2}{t_2}\\(b)\;t_2=-t_1-\large\frac{2}{t_1}\\(c)\;2t_1=-3t_2-\large\frac{2}{t_2}\\(d)\;4t_1=-t_2-\large\frac{3}{t_2}\end{array}$

Parametric equation of normal at $(at_1^2,2at_1)$ is
$y=-t_1x+2at_1+at_1^3$---------(1)
Since it meet the parabola again at $(at_2^2,2at_2)$
$2at_2=-at_1t_2^2+2at_1+at_1^3$
$2a(t_2-t_1)+at_1(t_2^2-t_1^2)=0$
$a(t_2-t_1)[2+t_1(t_2+t_1)]=0$
$a(t_2-t_1)\neq 0$
$t_1$ and $t_2$ are different.
$2+t_1(t_2+t_1)=0$
$t_2=-t_1-\large\frac{2}{t_1}$
Hence (b) is the correct answer.