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

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$t_{1} + t_{2} + t_{3} = t_{1} t_{2} t_{3}$

$t_{1} + t_{2} + t_{3} = 0$

$t_{1} + t_{2} + t_{3} = - 1$

$t_{1} t_{2} t_{3} = 1$

answer is C.

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Detailed Solution

The given points are collinear if

$\begin{aligned} |\begin{array}{lcc} t_{1} & 2 a t_{1} + a t_{1}^{3} & 1 \\ t_{2} & 2 a t_{2} + a t_{2}^{3} & 1 \\ t_{3} & 2 a t_{3} + a t_{3}^{3} & 1 \end{array}| = 0 \\ \Rightarrow |\begin{array}{lll} t_{1} & 2 t_{1} + t_{1}^{3} & 1 \\ t_{2} & 2 t_{2} + t_{2}^{3} & 1 \\ t_{3} & 2 t_{3} + t_{3}^{3} & 1 \end{array}| = 0 \\ \Rightarrow |\begin{array}{ccc} t_{1} & 2 t_{1} + t_{1}^{3} & 1 \\ t_{2} - t_{1} & 2 (t_{2} - t_{1}) + (t_{2}^{3} - t_{1}^{3}) & 0 \\ t_{3} - t_{1} & 2 (t_{3} - t_{1}) + (t_{3}^{3} - t_{1}^{3}) & 0 \end{array}| = 0 \end{aligned}$

Applying $R_{2} \to R_{2} - R_{1}$

$R_{3} \to R_{3} - R_{1}$ we get

$\Rightarrow (t_{2} - t_{1}) (t_{3} - t_{1}) |\begin{array}{ccc} t_{1} & 2 t_{1} + t_{1}^{3} & 1 \\ 1 & 2 + t_{2}^{2} + t_{1}^{2} + t_{2} t_{1} & 0 \\ 1 & 2 + t_{3}^{2} + t_{1}^{2} + t_{3} t_{1} & 0 \end{array}| = 0$

$\begin{array}{lll} \Rightarrow (t_{2} - t_{1}) (t_{3} - t_{1}) (t_{3} - t_{2}) (t_{3} + t_{2} + t_{1}) = 0 \\ \Rightarrow t_{1} + t_{2} + t_{3} = 0 ⌊∵ t_{1} \neq t_{2} \neq t_{3}] \end{array}$