If t1, t2 and t3 are distinct, the points (t1, 2 a t1, a t13),(t2, 2 a t2, a t23)(t3, 2 a t3, a t33) are collinear, if

Question

If t1, t2 and t3 are distinct, the points (t1, 2 a t1, a t13),(t2, 2 a t2, a t23)(t3, 2 a t3, a t33) are collinear, if (A)  t 1 t2 t3=1  (B)  t 1 + t2 + t3 = t1 t2 t3  (C) t 1 + t2 + t 3 = 0  (D)  t 1 + t2 + t3 = -1

Tardigrade Admin · Accepted Answer

The given points are collinear, if |t1 2 a t1+a t13 1 t2 2 a t2+a t23 1 t3 2 a t3+a t33 1|=0 ⇒ a|t1 2 t1+t13 1 t2 2 t2+t23 1 t3 2 t3+t33 1|=0 Applying R2 arrow R2-R1, R3 arrow R3-R1, we get |t1 2 t1+t13 1 t2-t1 2(t2-t1)+(t23-t13) 0 t3-t1 2(t3-t1)+(t33-t13) 0| ⇒ (t2-t1)(t3-t1) |t1 2 t1+t13 1 1 2+t22+t12+t2 t1 0 1 2+t32+t12+t3 t1 0|=0 ⇒ (t2-t1)(t3-t1)(t3- t2)(t3+t2+t1)=0 ⇒ t1+t2+t3=0[∵ t1 ≠ t2 ≠ t3]

Q. If $t_{1}, t_{2}$ and $t_{3}$ are distinct, the points $(t_{1}, 2 a t_{1}, a t_{1}^{3}), (t_{2}, 2 a t_{2}, a t_{2}^{3}) (t_{3}, 2 a t_{3}, a t_{3}^{3})$ are collinear, if

$t_{1} t_{2} t_{3} = 1$

$t_{1} + t_{2} + t_{3} = t_{1} t_{2} t_{3}$

$t_{1} + t_{2} + t_{3} = 0$

$t_{1} + t_{2} + t_{3} = - 1$

Q. If t1​,t2​ and t3​ are distinct, the points (t1​,2at1​,at13​),(t2​,2at2​,at23​)(t3​,2at3​,at33​) are collinear, if

t1​t2​t3​=1

t1​+t2​+t3​=t1​t2​t3​

t1​+t2​+t3​=0

t1​+t2​+t3​=−1

Q. If $t_{1}, t_{2}$ and $t_{3}$ are distinct, the points $(t_{1}, 2 a t_{1}, a t_{1}^{3}), (t_{2}, 2 a t_{2}, a t_{2}^{3}) (t_{3}, 2 a t_{3}, a t_{3}^{3})$ are collinear, if

$t_{1} t_{2} t_{3} = 1$

$t_{1} + t_{2} + t_{3} = t_{1} t_{2} t_{3}$

$t_{1} + t_{2} + t_{3} = 0$

$t_{1} + t_{2} + t_{3} = - 1$