Q103 / 200 · Manipal (MET) 2010 · Mathematics

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
    $t_1+t_2+t_3=t_1 t_2 t_3$
  • Correct answer
    C
    $t_1+t_2+t_3=0$
  • D
    $t_1+t_2+t_3=-1$

Solution

The given points are collinear, if $\left|\begin{array}{lll}t_1 & 2 a t_1+a t_1^3 & 1 \\ t_2 & 2 a t_2+a t_2^3 & 1 \\ t_3 & 2 a t_3+a t_3^3 & 1\end{array}\right|=0$ $\Rightarrow a\left|\begin{array}{lll}t_1 & 2 t_1+t_1^3 & 1 \\ t_2 & 2 t_2+t_2^3 & 1 \\ t_3 & 2 t_3+t_3^3 & 1\end{array}\right|=0$ Applying $R_2 \rightarrow R_2-R_1, R_3 \rightarrow R_3-R_1$, we get $\left|\begin{array}{ccc}t_1 & 2 t_1+t_1^3 & 1 \\ t_2-t_1 & 2\left(t_2-t_1\right)+\left(t_2^3-t_1^3\right) & 0 \\ t_3-t_1 & 2\left(t_3-t_1\right)+\left(t_3^3-t_1^3\right) & 0\end{array}\right|=0$ $\Rightarrow\left(t_2-t_1\right)\left(t_3-t_1\right)$ $\left|\begin{array}{ccc}t_1 & 2 t_1+t_1^3 & 1 \\ 1 & 2+t_2^2+t_1^2+t_2 t_1 & 0 \\ 1 & 2+t_3^2+t_1^2+t_3 t_1 & 0\end{array}\right|=0$ $\Rightarrow\left(t_2-t_1\right)\left(t_3-t_1\right)\left(t_3-t_2\right)\left(t_3+t_2+t_1\right)=0$ $\Rightarrow t_1+t_2+t_3=0 \quad\left[\because t_1 \neq t_2 \neq t_3\right]$

Hence, the correct answer is option: 'C'.

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