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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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a

t1t2t3=1

b

t1+t2+t3=t1t2t3

c

t1+t2+t3=0

d

t1+t2+t3=−1

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detailed solution

Correct option is C

The given points are collinear ift12at1+at131t22at2+at231t32at3+at331=0⇒t12t1+t131t22t2+t231t32t3+t331=0⇒t12t1+t131t2−t12t2−t1+t23−t130t3−t12t3−t1+t33−t130=0Applying R2→R2−R1R3→R3−R1 we get⇒ t2−t1t3−t1t12t1+t13112+t22+t12+t2t1012+t32+t12+t3t10=0⇒ t2−t1t3−t1t3−t2t3+t2+t1=0 ⇒ t1+t2+t3=0 ∵t1≠t2≠t3

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