Question

Let $A \left( at _1^2, 2 a t _1\right) ; B \left( a t_2^2, 2 a t _2\right) ; C \left( at _3^2, 2 a t _3\right)$ be three points on the parabola $y ^2=4 ax$. If the orthocentre of the triangle $A B C$ is at the focus, then find $\left|t_1+t_2+t_3+t_1 t_2 t_3+t_1 t_2+t_2 t_3+t_3 t_1\right|$.

Answer 1

$\frac{2}{ t _1+ t _2} \times \frac{2 at _3}{ a \left( t _3^2-1\right)}=-1$
$\frac{2}{t_1+t_2}=\frac{1-t_3^2}{2 t_3} $
$t_1+t_2=\frac{4 t_3}{1-t_3^2}$
$t_1+t_2+t_3=\frac{4 t_3}{1-t_3^2}+t_3=\frac{4 t_3+t_3-t_3^3}{1-t_3^2}=\frac{5 t_3-t_3^3}{1-t_3^2}$
If $t_1+t_2+t_3=u$ then $u=\frac{5 t_3-t_3^3}{1-t_3^2}$ .....(1)
The Equation (1) suggests that $t _3$ is a root of
$t^3-u t^2-5 t+u=0$ ....(2)
by symmetry $t_1$ and $t_2$ are also the root of (2). Hence
$t_1 + t_2 + t_3 = u$
and $t_1 t_2 t_3=-u$
Hence $t_1+t_2+t_3+t_1 t_2 t_3=0$
also $t_1 t_2+t_2 t_3+t_3 t_1=-5$
Hence $\left| t _1+ t _2+ t _3+ t _1 t _2 t _3+ t _1 t _2+ t _2 t _3+ t _3 t _1\right|=5$