If t1, t2 and t3 are distinct, the points t1, 2at1+at13 t2,2at2+at23 and t3,2at3+at33 are collinear if

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