If the normal at t₁ on the parabola y²= 4ax meets the curve again at a point t₂ then show that ${\mathrm{t}}_2=-{\mathrm{t}}_1-\frac{2}{{\mathrm{t}}_1}$

The equation to the normal at t₁ on the parabola y² = 4ax is
$\mathrm{y}+{\mathrm{xt}}_1=2{\mathrm{at}}_1+{\mathrm{at}}_{1 }^3 ...\left(1\right)$
If (1) passes through $\left({\mathrm{at}}_2^2,2{\mathrm{at}}_2\right)$ then
$\begin{array}{l}2{\mathrm{at}}_2+{\mathrm{at}}_2^2{\mathrm{t}}_1=2{\mathrm{at}}_1+{\mathrm{at}}_1^3\\ \Rightarrow 2\left({\mathrm{t}}_2-{\mathrm{t}}_1\right)={\mathrm{t}}_1\left({\mathrm{t}}_1^2-{\mathrm{t}}_2^2\right)\\ -2={\mathrm{t}}_1\left({\mathrm{t}}_1+{\mathrm{t}}_2\right)\Rightarrow -\frac{2}{{\mathrm{t}}_1}={\mathrm{t}}_1+{\mathrm{t}}_2\\ \Rightarrow {\mathrm{t}}_2=-{\mathrm{t}}_1-\frac{2}{{\mathrm{t}}_1}\end{array}$