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JEE MainMathematicsEasyStraight Lines

If the three distinct points
(ti,2ati+ati3)\left( t_{i},2{at}_{i} + {at}_{i}^{3} \right)for i=1, 2, 3 are
collinear then the sum of the abscissae of the points is;fn rhu fHkUu&fHkUu fcUnq
(ti,2ati+ati3)\left( t_{i},2{at}_{i} + {at}_{i}^{3} \right)] i=1, 2, 3 ds fy,
rc fcUnqvksa

Show answer and solution
$\left| \begin{matrix}
t_{1}\ \ \ \ 2at_{1} + at_{1}^{3}\ \ \ \ 1 \\
t_{2}\ \ \ \ 2at_{2} + at_{2}^{3}\ \ \ \ 1 \\
t_{3}\ \ \ \ 2at_{3} + at_{3}^{3}\ \ \ \ 1
\end{matrix} \right| = 0$
$\Rightarrow a\left| \begin{matrix}
t_{1} & t_{1}^{3} & 1 \\
t_{2} & t_{3}^{2} & 1 \\
t_{3} & t_{3}^{3} & 1
\end{matrix} \right| = 0$
⇒(t1−t2)(t2−t3)(t3−t1)(t1+t2+t3)=0⇒t1+t2+t3=0\Rightarrow \left( t_{1} - t_{2} \right)\left( t_{2} - t_{3} \right)\left( t_{3} - t_{1} \right)\left( t_{1} + t_{2} + t_{3} \right) = 0 \Rightarrow t_{1} + t_{2} + t_{3} = 0

Source: Grass-G / M-25 Mathematics Paper 7 Jan.docx

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