Download App

Complex Numbers — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSComplex Numbers4 Marks
Question
If $z_1, z_2$ are two complex numbers such that $|\text{z}_1|=|\text{z}_2|$ and $\text{arg(z}_1)+\text{arg(z}_2)=\pi,$ then show that $\text{z}_1=-\bar{\text{z}}_2.$
✓

Answer

$|\text{z}_1|=|\text{z}_2|$ Let $\text{arg(z}_1)=\theta$
$\therefore\text{arg(z}_2)=\pi-\theta$ In polar form, $\text{z}_1=|\text{z}_1|(\cos\theta+\text{i}\sin\theta) \ ...(\text{i})$
$\text{z}_2=|\text{z}_2|\big(\cos(\pi-\theta)+\text{i}\sin(\pi-\theta\big)$
$=|\text{z}_2|\big(-\cos\theta+\text{i}\sin\theta\big)$
$=-|\text{z}_2|\big(\cos\theta-\text{i}\sin\theta\big)$ Finding conjugate of $\bar{\text{z}}_2=-|\text{z}_2|\big(\cos\theta-\text{i}\sin\theta\big) \ ...(\text{ii})$ (i), (ii) is equal to $\frac{\text{z}_1}{\text{z}_2}=-\frac{|\text{z}_1|(\cos\theta+\text{i}\sin\theta)}{|\text{z}_2|(\cos\theta+\text{i}\sin\theta)}$
$\frac{\text{z}_1}{\text{z}_2}=-\frac{|\text{z}_1|}{|\text{z}_2|} \ [\because|\text{z}_1|=|\text{z}_2|]$
$\frac{\text{z}_1}{\text{z}_2}=-1$
$\text{z}_1=\bar{-\text{z}}_2$ Hence proved.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "(Each question 4 marks)" from Complex Numbers→Complex Numbers — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the number of words formed by permuting all the letters of the following words: SERIES.
→
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): $\frac{\sin{(\text{x+a)}}}{\cos\text{x}}$
→
Solve the following systems of inequations graphically: $2\text{x} + \text{y} \geq 8, \text{x} + 2\text{y} \geq 8,\text{ x} +\text{ y}\leq 6$
→
Find the equation of the line joining the point (3, 5) to the point of intersection of the lines 4x + y - 1 = 0 and 7x - 3y - 35 = 0.
→
Let f and g be two real functions defined by $\text{f(x)}=\sqrt{\text{x}+1}$ and $\text{g(x)}=\sqrt{9-\text{x}^2}$ Then describe the following functions: $\frac{\text{f}}{\text{g}}$
→
For any two sets of A and B, prove that:

$\text{A}'\cup\text{B}=\text{U}\Rightarrow\text{A}\subset\text{B.}$

→
Find the derivative of $x^2– 2$ at $x = 10$.
→
The $4^{th}$ and $7^{th}$ terms of G.P. are $\frac{1}{27}\text{ and }\frac{1}{729}$ respectively. Find the sum of n terms of the G.P.
→
Prove that: $\frac{1}{\cos\text{(x}-\text{b})\cos\text{(x}-\text{b)}}=\frac{\tan\text{(x}-\text{a)}-\tan\text{(x}-\text{b)}}{\sin\text{(a}-\text{b)}}$
→
Determine the number of 5 cards combinations out of a deck of 52 cards if at least one of the 5 cards has to be a king?
→