Question

Find the value of
$\left | \dfrac{ {| \overset - z| ^{2}} }{z. \overset - z} \right|$
​

Answer 1

Question :-

If z is a complex number, find the value of following :

$\sf \: \left | \dfrac{ {| \overset - z| ^{2}} }{z. \overset - z} \right|$

$\large\underline{\sf{Solution-}}$

Given expression is to find the value of,

$\rm :\longmapsto\:\left | \dfrac{ {| \overset - z| ^{2}} }{z. \overset - z} \right|$

Since, z is a complex number.

Let assume that

$\rm :\longmapsto\:z = x + iy$

So,

$\rm :\longmapsto\:\overset - z \: = \: x - iy$

Consider,

$\red{\rm :\longmapsto\:z\overset - z}$

$\rm \: = \:(x + iy)(x - iy)$

$\rm \: = \: {x}^{2} - {(iy)}^{2}$

$\rm \: = \: {x}^{2} - {i}^{2} {y}^{2}$

As, we know

$\boxed{ \tt{ \: {i}^{2} = - \: 1 \: }}$

So,

$\rm \: = \: {x}^{2} + {y}^{2}$

$\rm \implies\:\boxed{ \tt{ \: z\overset - z = {x}^{2} + {y}^{2} \: }}$

Now, Consider

$\red{\rm :\longmapsto\: |\overset - z| \: }$

$\rm \: = \: |x - iy|$

$\rm \: = \: \sqrt{ {x}^{2} + {y}^{2} }$

Thus,

$\rm \implies\:\boxed{ \tt{ \: { |\overset - z| }^{2} = {x}^{2} + {y}^{2} \: }}$

Now, Consider

$\red{\rm :\longmapsto\:\left | \dfrac{ {| \overset - z| ^{2}} }{z. \overset - z} \right|}$

$\rm \: = \: \left |\dfrac{ {x}^{2} + {y}^{2} }{ {x}^{2} + {y}^{2} } \right|$

$\rm \: = \:1$

Thus,

$\rm \implies\:\boxed{ \tt{ \: \left | \dfrac{ {| \overset - z| ^{2}} }{z. \overset - z} \right| = 1 \: }}$

More to know :-

$\boxed{ \tt{ \: |z| = |\overset - z| }}$

$\boxed{ \tt{ \: z\overset - z = { |z| }^{2} \: }}$

$\boxed{ \tt{ \: \overline{z_1 + z_2} \: = \overset - z_1 + \overset - z_2 }}$

$\boxed{ \tt{ \: \overline{z_1 - z_2} \: = \overset - z_1 - \overset - z_2 }}$

$\boxed{ \tt{ \: \overline{z_1 \times z_2} \: = \overset - z_1 \times \overset - z_2 }}$

$\boxed{ \tt{ \: \overline{z_1 \div z_2} \: = \overset - z_1 \div \overset - z_2 }}$

$\boxed{ \tt{ \: |z_1z_2| = |z_1| |z_2| \: }}$