Let z 1 and z 2 be two distinct complex numbers and let z = ( 1 - t ) z 1 + t z 2 for some real number t with 0 t 1 . If Arg ( w ) denotes the principal argument of a non-zero complex number w , then [2010]

Question

Let $z_{1}$ and $z_{2}$ be two distinct complex numbers and let $z = (1 - t) z_{1} + t z_{2}$ for some real number $t$ with $0 < t < 1$ . If $Arg (w)$ denotes the principal argument of a non-zero complex number $w$ , then [2010]

Smart Achievers · Accepted Answer

(1, 3, 4)

$Given: z = (1 - t) z_{1} + t z_{2}, where 0 < t < 1$

$\Rightarrow z = \frac{(1 - t) z_{1} + t z_{2}}{(1 - t) + t}$

$\Rightarrow z divides the join of z_{1} and z_{2} internally in the ratio t : (1 - t)$

$∴$ $z_{1}, z, z_{2} are collinear$

$\Rightarrow | z - z_{1} | + | z - z_{2} | = | z_{1} - z_{2} |$

$Also z = (1 - t) z_{1} + t z_{2}$

$\Rightarrow \frac{z - z_{1}}{z_{2} - z_{1}} = t, which is purely real number$

$∴$ $\arg (\frac{z - z_{1}}{z_{2} - z_{1}}) = 0 \Rightarrow \arg (z - z_{1}) = \arg (z_{2} - z_{1})$

$Also \frac{z - z_{1}}{z_{2} - z_{1}} = t \Rightarrow \frac{\bar{z} - {\bar{z}}_{1}}{{\bar{z}}_{2} - {\bar{z}}_{1}} = t$

$\Rightarrow \frac{z - z_{1}}{z_{2} - z_{1}} = \frac{\bar{z} - {\bar{z}}_{1}}{{\bar{z}}_{2} - {\bar{z}}_{1}}$

$\Rightarrow (z - z_{1}) ({\bar{z}}_{2} - {\bar{z}}_{1}) = (\bar{z} - {\bar{z}}_{1}) (z_{2} - z_{1})$

$\Rightarrow$ $| \begin{matrix} z - z_{1} & \bar{z} - {\bar{z}}_{1} \\ z_{2} - z_{1} & {\bar{z}}_{2} - {\bar{z}}_{1} \end{matrix} |$ $= 0$

Solution

Q.
Let $z_{1}$ and $z_{2}$ be two distinct complex numbers and let $z = (1 - t) z_{1} + t z_{2}$ for some real number $t$ with $0 < t < 1$ . If $Arg (w)$ denotes the principal argument of a non-zero complex number $w$ , then [2010]

1 $| z - z_{1} | + | z - z_{2} | = | z_{1} - z_{2} |$

2 $Arg (z - z_{1}) = Arg (z - z_{2})$

3 $| \begin{matrix} z - z_{1} & \bar{z} - \bar{z_{1}} \\ z_{2} - z_{1} & \bar{z_{2}} - \bar{z_{1}} \end{matrix} |$

4 $Arg (z - z_{1}) = Arg (z_{2} - z_{1})$

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Solution

Q. Let z1 and z2 be two distinct complex numbers and let z=(1-t)z1+tz2 for some real number t with 0<t<1. If Arg(w) denotes the principal argument of a non-zero complex number w, then [2010]

1 |z-z1|+|z-z2|=|z1-z2|

2 Arg(z-z1)=Arg(z-z2)

3 |z-z1z¯-z1¯z2-z1z2¯-z1¯|