Question

If the chord joining the points t1 and t2 to the parabola y2 = 4ax is normal to the parabola at t-th prove that t1(t1+ t2) =-2.

Answer 1

The equation of normal to the parabola

$y^2=4ax$ at the point $t_1\:(a{t_1}^2,2at_1)$ is

$\bf\,xt_1+y=a{t_1}^3+2at_1$

But it meets the parabola again at $t_2\:(a{t_2}^2,2at_2)$

Therefore, We have

$a{t_2}^2\,t_1+2at_2=a{t_1}^3+2at_1$

$a{t_2}^2\,t_1-a{t_1}^3=2at_1-2at_2$

$at_1({t_2}^2-{t_1}^2)=2a(t_1-t_2)$

$at_1({t_2}-{t_1})({t_2}+{t_1})=2a(t_1-t_2)$

$-at_1({t_1}-{t_2})({t_1}+{t_2})=2a(t_1-t_2)$

$-t_1({t_1}+{t_2})=2$

$\implies\,\boxed{\bf\,t_1({t_1}+{t_2})=-2}$