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# Let $(x_1,y_1)$ be the point of intersection of the tangent to the parabola $y^2=4ax$ at the points $t_1$ and $t_2$ . Then $x_1=a{t_1}{t_2}$ and $y_1=a(t_1+t_2)$

Prove that if  $(x_1,y_1)$ be the point of intersection of the tangent to the parabola $y^2=4ax$ at the points $t_1$ and $t_2$ . Then $x_1=a{t_1}{t_2}$ and $y_1=a(t_1+t_2)$

Let the equation of the two tangents at the points $A(t_1)$ and $B(t_2)$ be $yt_1=x+at_1^2$ and $yt_2=x+at_2^2$
Subtract both the equations $y (t_1-t_2)=a(t_1^2-t_2)y(t_1 -t_2) =a(t_1-t_2)$
Hence $y=a(t_1+t_2)$ Substituting in equal $a(t_1+t_2) t_1=x+at_1^2$
Therefore at $1^2 +t_1 t_2 =x +t_1^2$
Therefore $x=t_1t_2$