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

The given points are collinear, ift12at1+at131t22at2+at231t32at3+at331=0⇒at12t1+t131t22t2+t231t32t3+t331=0Applying R2→R2−R1,R3−R1,  we get                 t12t1+t121t2−t12(t2−t1)+(t23−t13)0t3−t12(t3−t1)+(t33−t13)0=0 ⇒(t2−t1)(t3−t1)t12t1+t13112+t22−t12+t2t1012+t32+t12+t3t10=0⇒(t2−t1)(t3−t1)(t3−t2)(t3+t2+t1)=0⇒t1+t2+t3=0