Question

If the points (a, 0), (at₁², 2at₁) and (at₂², 2at₂) are collinear, write the value of t₁ t₂.

 

Answer 1

For the points (a, 0), (at₁², 2at₁) and (at₂², 2at₂) to be collinear, the following condition has to be met:

\[\begin{vmatrix}a & 0 & 1 \\ a {t_1}^2 & 2a t_1 & 1 \\ a {t_2}^2 & 2a t_2 & 1\end{vmatrix} = 0\]
\[ \Rightarrow a\left( 2a t_1 - 2a t_2 \right) - 0 + 1\left( 2 a^2 {t_1}^2 t_2 - 2 a^2 t_1 {t_2}^2 \right) = 0\]
\[ \Rightarrow 2 a^2 \left( t_1 - t_2 \right) + 2 a^2 t_1 t_2 \left( t_1 - t_2 \right) = 0\]
\[ \Rightarrow 2 a^2 \left( t_1 - t_2 \right)\left( 1 + t_1 t_2 \right) = 0\]
\[\Rightarrow \left( t_1 - t_2 \right) = 0 or \left( 1 + t_1 t_2 \right) = 0 \left( a \neq 0 \right)\]
\[ \Rightarrow 1 + t_1 t_2 = 0 \left( \because t_1 \neq t_2 \right)\]
\[ \Rightarrow t_1 t_2 = - 1\]