Let z-1 and z-2 be two distinct complex numbers and let z-1-t-z-1 -tz-2 for some real number t with 0 lt-t lt-1- If arg- denotes the principal argument of a non-zero complex number - then-z-z-1-z-z-2-z-1-z-2-arg-z-z-1- - arg-z-z-2-z-z-1 amp- z-z-1-z-2-z-1 amp- z-2-z-1-0arg-z-z-1- - arg-z-2-z-1-

The correct options are A |z z_1|+|z+z_2|=|z_1 z_2| C z z_1 amp; z z_1 z_2 z_1 amp; z_2 z_1=0 D arg(z z_1) = arg(z_2 z_1)Given, z=(1 t)z_1 +tz_2/(1 t)t ...

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Byju's Answer

Standard XI

Mathematics

Section Formula Using Complex Numbers

Let z_1 and z...

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 $(ω)$ denotes the principal argument of a non-zero complex number $ω$ , then

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

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arg

(z - z_{1})

= arg

(z - z_{2})

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∣ ∣ ∣ \begin{matrix} z - z_{1} & ¯ ¯ ¯ z -_{1} z_{2} - z_{1} & _{2} -_{1} \end{matrix} ∣ ∣ ∣ = 0

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arg

(z - z_{1})

= arg

(z_{2} - z_{1})

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Solution

The correct options are
A $| z - z_{1} | + | z + z_{2} | = | z_{1} - z_{2} |$
C $∣ ∣ ∣ \begin{matrix} z - z_{1} & ¯ ¯ ¯ z -_{1} z_{2} - z_{1} & _{2} -_{1} \end{matrix} ∣ ∣ ∣ = 0$
D arg $(z - z_{1})$ = arg $(z_{2} - z_{1})$
Given, $z = \frac{(1 - t) z_{1} + t z_{2}}{(1 - t) t}$

Clearly, z divides $z_{1}$ and $z_{2}$ in the ratio of t : (1-t), 0<t<1
$\Rightarrow A P + B P = A B i . e . | z - z_{1} | + | z - z_{2} | = | z_{1} - z_{2} | \Rightarrow$ Option (a) is true.
And arg $(z - z_{1})$ =arg $(z_{2} - z)$
=arg $(z_{2} - z_{1})$
$\Rightarrow$ Option (b) is false and option (d) is true.
Also, arg $(z - z_{1})$ =arg $(z_{2} - z_{1})$
$\Rightarrow a r g (\frac{z - z_{1}}{z_{2} - x_{1}}) = 0$
$∴ \frac{z - z_{1}}{z_{2} - z_{1}}$ is purely real.
$∴ \frac{z - z_{1}}{z_{2} - z_{1}} = \frac{¯ z -_{1}}{_{2} -_{1}}$
or $∣ ∣ ∣ \begin{matrix} z - z_{1} & ¯ ¯ ¯ z -_{1} z_{2} - z_{1} & _{2} -_{1} \end{matrix} ∣ ∣ ∣ = 0$
$∴$ Option (c ) is correct. Hence, ( a, c, d ) is the correct option.

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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(ω) denotes the principal argument of a non-zero complex number ω, then

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 $(ω)$ denotes the principal argument of a non-zero complex number $ω$ , then