First slide

Geometry of complex numbers

Question

$If complex number z lies on the curve | z - (- 1 + i) | = 1, then the locus of the complex$
$number ω = \frac{z + i}{1 - i}, i = \sqrt{- 1} is a circle having$

Moderate

A

$centre at (- 3 / 2, 1 / 2) and radius \frac{1}{\sqrt{2}}$
B

$centre at (3 / 2, - 1 / 2) and radius \frac{1}{\sqrt{2}}$
C

$centre at (3 / 2, - 1 / 2) and radius \sqrt{2}$
D

$centre at (- 3 / 2, 1 / 2) and radius \sqrt{2}$

Solution

$\begin{array}{l} We have | z - (- 1 + i) | = 1 \\ \Rightarrow | z + 1 - i | = 1 \\ Now, ω = \frac{z + i}{1 - i} \\ \Rightarrow (1 - i) ω = z + i \\ a d d i n g - 2 i + 1 o n b o t h s i d e s \\ \Rightarrow (1 - i) ω - 2 i + 1 = z + 1 - i \\ \Rightarrow | (1 - i) ω - 2 i + 1 | = | z + 1 - i | \end{array}$
$\begin{array}{l} \Rightarrow | 1 - i | |ω + \frac{1 - 2 i}{1 - i}| = 1 \Rightarrow |ω + \frac{(1 - 2 i) (1 + i)}{(1 + i) (1 - i)}| = \frac{1}{\sqrt{2}} \\ \Rightarrow |ω - \frac{- 3 + i}{2}| = \frac{1}{\sqrt{2}} \end{array}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App