if the cord joining the points t1 and t2 to the parabola y^2 =4ax is normal to the parabola at t1 then prove that t1(t1+t2)=-2

The given parabola is y^2 = 4ax.

The equation of the normal at any point (at^2, 2at) on the parabola is y = -tx + 2at + at^3.

The equation of the chord joining the points t1 and t2 on the parabola is y = a(t1 + t2)x - at1t2.

Given that the chord is normal to the parabola at t1, the equations of the normal and the chord must be the same.

Therefore, we have:

-t1x + 2at1 + at1^3 = a(t1 + t2)x - at1t2.

Solving this equation for x gives:

x = (2at1 + at1^3 + at1t2) / (t1 + a(t1 + t2)).

But we know that x = at1^2 for any point on the parabola.

So, we have:

at1^2 = (2at1 + at1^3 + at1t2) / (t1 + a(t1 + t2)).

Solving this equation for t1(t1 + t2) gives:

t1(t1 + t2) = -2.

So, we have proved that if the chord joining the points t1 and t2 to the parabola y^2 = 4ax is normal to the parabola at t1, then t1(t1 + t2) = -2.