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Mathematics Question on Complex Numbers and Quadratic Equations

Let z1z_1 and z2z_2 be two distinct complex numbers and let z=(1−t)z1+tz2z = (1 - t) z_1 + tz_2 for some real number t with 0<t<10 < t < 1. If arg(w)arg (w) denotes the principal argument of a non-zero complex num ber w, then

A

∣z−z1∣+∣z−z2∣=∣z1−z2∣|z-z_1|+|z-z_2|=|z_1-z_2|

B

arg(z-z2)z_2) = arg(z-z2)z_2)

C

∣z−z1z‾−z1‾ z2−z1z2‾−z1‾∣=0\begin {vmatrix} z-z_1 & \overline{z}-\overline{z_1} \\\ z_2-z_1 & \overline{z_2}-\overline{z_1} \end {vmatrix} =0

D

arg(z-z1)=arg(z2−z1)z_1)=arg(z_2-z_1)

Answer

arg(z-z1)=arg(z2−z1)z_1)=arg(z_2-z_1)

Explanation

Solution

Given , z=(1−t)z1+tz2(1−t)+t\frac{(1-t)z_1+tz_2}{(1-t)+t} Clearly, 2 divides z1_1and z2_2 in the ratio of t : (1 - 1), 0 < t < 1 ⇒   AP+BP+AB  i.e.,∣z−z1∣+∣z−z2∣=∣z1−z2∣\Rightarrow \ \ \ AP+BP+AB \ \ i.e., |z-z_1|+|z-z_2|=|z_1-z_2| ⇒  \Rightarrow \ \ Option (a) is true. and a rg (z - z1z_1) =arg(z2z_2-z ) =arg(z2z_2- z1z_1) ⇒\Rightarrow Option (b) is false and option (d) is true. Also, arg(z-z1)=arg(z2−z)=arg(z2−z1)z_1)=arg(z_2-z)=arg(z_2-z_1) →  arg(z−z1z2−z1)=0\rightarrow \ \ arg\bigg( \frac{z-z_1}{z_2-z_1}\bigg)=0 ∴z−z1z2−z1\therefore \frac{z-z_1}{z_2-z_1} is purely real. ⇒  z−z1z2−z1=z‾−z1‾z2‾−z1‾\Rightarrow \ \ \frac{z-z_1}{z_2-z_1}=\frac{\overline{z}-\overline{z_1}}{\overline{z_2}-\overline{z_1}} or ∣z−z1z‾−z1‾ z2−z1z2‾−z1‾∣=0\begin {vmatrix} z-z_1 & \overline{z}-\overline{z_1} \\\ z_2-z_1 & \overline{z_2}-\overline{z_1} \end {vmatrix}=0 ∴\therefore Option (c) is correct. Hence, (a, c, d) is the correct option.