Question

If $t_{1}, t_{2}$ and $t_{3}$ are distinct, the points $\left(t_{1}, 2 a t_{1}, a t_{1}^{3}\right),\left(t_{2}, 2 a t_{2}, a t_{2}^{3}\right)\left(t_{3}, 2 a t_{3}, a t_{3}^{3}\right)$ are collinear, if

• A. $ t _1\, t_2 \,t_3=1 $
• B. $ t _1 \,+\,t_2\,+\,t_3\,=\,t_1\, t_2\,t_3 $
• C. $t _1 + \,t_2\,+ \,t _3 \,= 0 $
• D. $ t _1\,+\, t_2\,+\, t_3\,= -1 $

Answer 1

Answer: (C)

The given points are collinear, if
$\begin{vmatrix}t_{1} & 2 a t_{1}+a t_{1}^{3} & 1 \\ t_{2} & 2 a t_{2}+a t_{2}^{3} & 1 \\ t_{3} & 2 a t_{3}+a t_{3}^{3} & 1\end{vmatrix}=0$
$\Rightarrow a\begin{vmatrix}t_{1} & 2 t_{1}+t_{1}^{3} & 1 \\ t_{2} & 2 t_{2}+t_{2}^{3} & 1 \\ t_{3} & 2 t_{3}+t_{3}^{3} & 1\end{vmatrix}=0$
Applying $R_{2} \rightarrow R_{2}-R_{1}, R_{3} \rightarrow R_{3}-R_{1}$, we get
$\begin{vmatrix}t_{1} & 2 t_{1}+t_{1}^{3} & 1 \\ t_{2}-t_{1} & 2\left(t_{2}-t_{1}\right)+\left(t_{2}^{3}-t_{1}^{3}\right) & 0 \\ t_{3}-t_{1} & 2\left(t_{3}-t_{1}\right)+\left(t_{3}^{3}-t_{1}^{3}\right) & 0\end{vmatrix}$
$\Rightarrow \left(t_{2}-t_{1}\right)\left(t_{3}-t_{1}\right)$
$\begin{vmatrix}t_{1} & 2 t_{1}+t_{1}^{3} & 1 \\ 1 & 2+t_{2}^{2}+t_{1}^{2}+t_{2} t_{1} & 0 \\ 1 & 2+t_{3}^{2}+t_{1}^{2}+t_{3} t_{1} & 0\end{vmatrix}=0$
$\Rightarrow \left(t_{2}-t_{1}\right)\left(t_{3}-t_{1}\right)\left(t_{3}- t_{2}\right)\left(t_{3}+t_{2}+t_{1}\right)=0 $
$\Rightarrow t_{1}+t_{2}+t_{3}=0\left[\because t_{1} \neq t_{2} \neq t_{3}\right]$