Let z1 and z2 be two distinct complex numbers and let z = (1 - t) z1 + tz2 for some real number t with 0

Question

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 (A) |z-z1|+|z-z2|=|z1-z2| (B) arg(z-z2) = arg(z-z2) (C) | z-z1 overlinez- overlinez1 z2-z1 overlinez2- overlinez1 | =0 (D) arg(z-z1)=arg(z2-z1)

Tardigrade Admin · Accepted Answer

Given , z=((1-t)z1+tz2/(1-t)+t) <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/chemistry/53ad65485b31d229228e0207acc21a69-.png" /> Clearly, z divides z1 and z2 in the ratio of t: (1 - 1), 0 < t < 1 ⇒ AP+BP+AB i.e., |z-z1|+|z-z2|=|z1-z2| ⇒ Option (a) is true. and arg (z - z1) =arg(z2 - z ) =arg(z2 - z1) ⇒ Option (b) is false and option (d) is true. Also, arg(z-z1)=arg(z2-z)=arg(z2-z1) arrow arg( (z-z1/z2-z1))=0 ∴ (z-z1/z2-z1) is purely real. ⇒ (z-z1/z2-z1)=( overlinez- overlinez1/ overlinez2- overlinez1) or | z-z1 overlinez- overlinez1 z2-z1 overlinez2- overlinez1 |=0 ∴ Option (c) is correct. Hence, (a, c, d) is the correct option.

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

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

$a r g (z - z_{2}) = a r g (z - z_{2})$

$∣ ∣ z - z_{1} z_{2} - z_{1} \overline{z} - \overline{z_{1}} \overline{z_{2}} - \overline{z_{1}} ∣ ∣ = 0$

$a r g (z - z_{1}) = a r g (z_{2} - z_{1})$

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

∣z−z1​∣+∣z−z2​∣=∣z1​−z2​∣

arg(z−z2​)=arg(z−z2​)

∣∣​z−z1​z2​−z1​​z−z1​​z2​​−z1​​​∣∣​=0

arg(z−z1​)=arg(z2​−z1​)

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

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

$a r g (z - z_{2}) = a r g (z - z_{2})$

$∣ ∣ z - z_{1} z_{2} - z_{1} \overline{z} - \overline{z_{1}} \overline{z_{2}} - \overline{z_{1}} ∣ ∣ = 0$

$a r g (z - z_{1}) = a r g (z_{2} - z_{1})$