Question

plz answer it guys l need help​

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

We are aware of the formula on complex number

1.

$\sf{ \:If \: \omega \: \: is \: cube \: root \: of \: unity \: then \: }$

${ \omega}^{3} = 1 \: \: and \: \: \:1 +\omega + {\omega}^{2} = 0$

2.

$\sf \: { If \: z = x + iy \: \: then \: |z| = \sqrt{ {x}^{2} + {y}^{2} } \: }$

TO DETERMINE

1. The value of

${ \omega}^{18}$

2. The value of

${\omega}^{ - 105}$

3.

The Cartesian coordinates of locus of z if

$|z - 3| = 2$

EVALUATION

1.

${ \omega}^{18} = { ( { \omega}^{3} )}^{6} = {(1)}^{6} = 1$

2.

${ \omega}^{ - 105} = {({ \omega}^{3}) }^{ - 35} = {(1)}^{ - 35} = 1$

3.

Let

$z = x + iy$

So

$|z - 3| = 2$

$\implies \: |x + iy - 3| = 2$

$\implies \: |(x - 3) + iy | = 2$

$\implies \: \sqrt{ {(x - 3)}^{2} + {y}^{2} } = 2$

$\implies \: {(x - 3)}^{2} + {y}^{2} = {2 }^{2}$

$\implies \: {(x - 3)}^{2} + {y}^{2} = 4$

Which is in Cartesian Coordinate system

The above equation represents a CIRCLE of radius 2 unit with centre at ( 3 , 0 )