If A(z1), B(z2) and C(z3) are collinear points then

Question

If A(z1), B(z2) and C(z3) are collinear points then (A) (z3-z1)( barz2- barz1)=( barz3- barz1)(z2-z1) (B) z3-z1= barz2- barz1 (C) (z3-z1)( barz3- barz1)=(z2-z1)( barz2- barz1) (D) None of these

Tardigrade Admin · Accepted Answer

If A(z1), B(z2) and C(z3) are collinear, then arg ((z3-z1/z2-z1))=0, π ⇒ (z3-z1/z2-z1) is purely real ⇒ (z3-z1/z2-z1)=( barz3- barz1/ barz2- barz1) ⇒ (z3-z1)( barz2- barz1) =(z2-z1)( barz3- barz1)

Q. If $A (z_{1}), B (z_{2})$ and $C (z_{3})$ are collinear points then

$(z_{3} - z_{1}) (\overset{z}{ˉ}_{2} - \overset{z}{ˉ}_{1}) = (\overset{z}{ˉ}_{3} - \overset{z}{ˉ}_{1}) (z_{2} - z_{1})$

$z_{3} - z_{1} = \overset{z}{ˉ}_{2} - \overset{z}{ˉ}_{1}$

$(z_{3} - z_{1}) (\overset{z}{ˉ}_{3} - \overset{z}{ˉ}_{1}) = (z_{2} - z_{1}) (\overset{z}{ˉ}_{2} - \overset{z}{ˉ}_{1})$

None of these

Q. If A(z1​),B(z2​) and C(z3​) are collinear points then

(z3​−z1​)(zˉ2​−zˉ1​)=(zˉ3​−zˉ1​)(z2​−z1​)

z3​−z1​=zˉ2​−zˉ1​

(z3​−z1​)(zˉ3​−zˉ1​)=(z2​−z1​)(zˉ2​−zˉ1​)

None of these

If A(z1​),B(z2​) and C(z3​) are collinear, then arg(z2​−z1​z3​−z1​​)=0,π ⇒z2​−z1​z3​−z1​​ is purely real ⇒z2​−z1​z3​−z1​​=zˉ2​−zˉ1​zˉ3​−zˉ1​​ ⇒(z3​−z1​)(zˉ2​−zˉ1​) =(z2​−z1​)(zˉ3​−zˉ1​)

Q. If $A (z_{1}), B (z_{2})$ and $C (z_{3})$ are collinear points then

$(z_{3} - z_{1}) (\overset{z}{ˉ}_{2} - \overset{z}{ˉ}_{1}) = (\overset{z}{ˉ}_{3} - \overset{z}{ˉ}_{1}) (z_{2} - z_{1})$

$z_{3} - z_{1} = \overset{z}{ˉ}_{2} - \overset{z}{ˉ}_{1}$

$(z_{3} - z_{1}) (\overset{z}{ˉ}_{3} - \overset{z}{ˉ}_{1}) = (z_{2} - z_{1}) (\overset{z}{ˉ}_{2} - \overset{z}{ˉ}_{1})$