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

Hi Rohit,

y² = 4ax, will give dy/dx = 2a/y.

Hence, slope of tangent at t₁ is 2a/2at₁ = 1/t₁.

So slope of normal at point t1 is -t₁.

Also slope of line joining t₁ and t₂ is (2at₂ - 2at₁ / at₂² - at₁²), which is equal to 2/(t₁+t₂).

The two slopes are the same. Hence

2/(t₁+t₂) = -t₁, or t₁(t₁+t₂) = -2.

That solves the problem.

Hope that helps.

All the best,

Regards,

Ashwin (IIT Madras)