Let z1 and z2 be two distinct complex numbers and let z=(1−λ)z1+λz2 for some real number λ(≠1) .If arg(z) denotes the principal argument of a non-zero complex number z then which of the following is /are TRUE.

z=(1−λ)z1+λz2(1−λ)+λ ⇒ z divides z1z2¯ in the ratio λ:1−λ internally⇒z1,z,z2 are collinear ⇒|z−z1|+|z−z2|=|z1−z2| arg(z−z1z2−z1)=0 ∴z−z1z2−z1=z¯−z¯1z2−z1

Let z1 and z2 be two distinct complex numbers and let z=(1−λ)z1+λz2 for some real number λ(≠1) .If arg(z) denotes the principal argument of a non-zero complex number z then which of the following is /are TRUE.

z=(1−λ)z1+λz2(1−λ)+λ ⇒ z divides z1z2¯ in the ratio λ:1−λ internally⇒z1,z,z2 are collinear ⇒|z−z1|+|z−z2|=|z1−z2| arg(z−z1z2−z1)=0 ∴z−z1z2−z1=z¯−z¯1z2−z1

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Let $z_{1}$ and $z_{2}$ be two distinct complex numbers and let $z = (1 - λ) z_{1} + λ z_{2}$ for some real number $λ (\neq 1)$ .If arg(z) denotes the principal argument of a non-zero complex number z then which of the following is /are TRUE.

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If $0 < λ < 1$ then $| z - z_{1} | + | z - z_{2} | = | z_{1} - z_{2} |$

If $λ > 1$ then locus of z is a circle

If $0 < λ < 1$ then $| \begin{matrix} z - z_{1} & \bar{z} - {\bar{z}}_{1} \\ z_{2} - z_{1} & {\bar{z}}_{2} - {\bar{z}}_{1} \end{matrix} | = 0$

If $0 < λ < 1$ then $\arg (z - z_{1}) = \arg (z - z_{2})$

answer is A, B, C.

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Detailed Solution

$z = \frac{(1 - λ) z_{1} + λ z_{2}}{(1 - λ) + λ}$
$\Rightarrow$ z divides $\bar{z_{1} z_{2}}$ in the ratio $λ : 1 - λ$ internally
$\Rightarrow z_{1}, z, z_{2}$ are collinear
$\Rightarrow | z - z_{1} | + | z - z_{2} | = | z_{1} - z_{2} | \arg (\frac{z - z_{1}}{z_{2} - z_{1}}) = 0 ∴ \frac{z - z_{1}}{z_{2} - z_{1}} = \frac{\bar{z} - {\bar{z}}_{1}}{z_{2} - z_{1}}$