The three distinct points A(at12, 2at1), B(at22, 2at2) and C(0, a) (where a is a real number) are collinear, if

Question

The three distinct points A(at12, 2at1), B(at22, 2at2) and C(0, a) (where a is a real number) are collinear, if (A)  t1t2=-1  (B)  t1t2=1  (C)  2t1t2=t1+t2  (D)  t1+t2=a

Tardigrade Admin · Accepted Answer

If there points A(at12,2at1), B(at22, 2at2) and C(0,a), collinear, if | beginmatrix at12 2at1 1 at22 2at2 1 0 a 1 endmatrix |=0 Use operation; R2→ R2-R1,R3→ R3-R1 | beginmatrix at12 2at1 1 a(t22-t12) 2a(t2-t1) 0 -at12 a-2at1 0 endmatrix |=0 Expand with respect to C3 a(t2-t1) (t2+t1) (a-2at1) +2a2t12(t2-t1)=0 ⇒ a(t2-t1) (t1+t2) (a-2at1)+2at12 =0 ⇒ a(t2-t1) at1+at2-2at12 -2at1t2+2at12 =0 ⇒ a(t2-t1) (at1+at2-2at1t2)=0 ⇒ a2(t2-t1) (t1+t2-2t1+t2)=0 ⇒ t2-t1=0 or t1+t2-2t1t2=0 ⇒ t1=t2 or t1=t2

Q. The three distinct points $A (a t_{1}^{2}, 2 a t_{1}), B (a t_{2}^{2}, 2 a t_{2})$ and $C (0, a)$ (where $a$ is a real number) are collinear, if

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

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

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

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

Q. The three distinct points A(at12​,2at1​),B(at22​,2at2​) and C(0,a) (where a is a real number) are collinear, if

t1​t2​=−1

t1​t2​=1

2t1​t2​=t1​+t2​

t1​+t2​=a

Q. The three distinct points $A (a t_{1}^{2}, 2 a t_{1}), B (a t_{2}^{2}, 2 a t_{2})$ and $C (0, a)$ (where $a$ is a real number) are collinear, if

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

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

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

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