Question

If the normal at the points (at₁² , 2at₁) and (at₂² , 2at₂) meet on the parabola y² = 4ax, then t₁ t ₂ =2.

Answer 1

Let the equation of normal at (at₁² ,2at₁) and (at₂² , 2at₂) be

y = –t₁ x+2at₁ +at₁³

and y = – t₂ x+2at₂ +at₂³

meet the parabola y² = 4ax at (at₃² ,2at₃) then

t₃ = –t₁-\(\frac{2}{t_1}\) and t₃ = -t₂-\(\frac{2}{t_2}\)

∴ – t₁ –\(\frac{2}{t_1}\) =– t₂-\(\frac{2}{t_2}\)

t ₂ –t₁ = 2 \(\left(\frac{1}{t_1}\frac{1}{t_2}\right)\)⇒ t₂-t₁=2\(\left(\frac{t_2-t_1}{t_1t_2}\right)\)

⇒ t₁ t ₂ = 2