Question

if. Z1 and Z2 are two complex number such that | Z1 | = | Z2 | + | Z1 - Z2 | then. a) Im[ Z1/Z2 ] = 0. b)Re[ Z1/Z2 ] = 0. c) Im[ Z1/Z2 ] = Re[ Z1/Z2 ] d) none of these​

Answer 1

if. Z1 and Z2 are two complex number such that | Z1 | = | Z2 | + | Z1 - Z2 | then. a) Im[ Z1/Z2 ] = 0. b)Re[ Z1/Z2 ] = 0. c) Im[ Z1/Z2 ] = Re[ Z1/Z2 ] d) none of these

I ALSO WANT THE ANSWER OF SAME QUESTION

Answer 2

✪ANSWER✪

• $\boxed{a) Im[\frac{Z₁}{Z₂}] = 0}$

᯽FORMULA᯽

For any complex number z,

$1. \: z \bar{z} \: = { |z| }^{2}$

$2. \:z + \bar{z} = 2Re(z)$

$3. \:Re(z) = |z| ⇒ Im(z) = 0$

$4. \:|z| = |\bar{z}|$

(Try proving them by taking $z=x+iy$)

★EXPLAINATION★

Here

$|Z₁| = |Z₂| + |Z₁- Z₂|$

$⇒|Z₁| - |Z₂| = |Z₁- Z₂|$

$⇒(|Z₁| - |Z₂|)² = |Z₁- Z₂|²$

$⇒|Z₁|² - 2|Z₁||Z₂| + |Z₂|²$

$\:\:\:\:\:\:= |Z₁- Z₂|²$

Using formula (1)

$⇒|Z₁|² - 2|Z₁||Z₂| + |Z₂|²$

$\:\:\:\:\:\:= (Z₁- Z₂)(\bar{Z₁} - \bar{Z₂})$

$⇒|Z₁|² - 2|Z₁||Z₂| + |Z₂|²$

$\:\:\:\:\:\:= Z₁\bar{Z₁} - Z₁\bar{Z₂}- Z₂\bar{Z₁} + Z₂\bar{Z₂}$

Using formula (1)

$⇒|Z₁|² - 2|Z₁||Z₂| + |Z₂|²$

$\:\:\:\:\:\: = |Z₁|² - (Z₁\bar{Z₂} + \bar{Z₁}Z₂) + |Z₂|²$

$⇒2|Z₁||Z₂| = Z₁\bar{Z₂} + \bar{Z₁}Z₂$

Using formula (2)

$⇒2|Z₁||Z₂| = 2Re(Z₁\bar{Z₂})$

Using formula (4)

$⇒|Z₁||\bar{Z₂}| = Re(Z₁\bar{Z₂})$

$⇒|Z₁\bar{Z₂}| = Re(Z₁\bar{Z₂})$

Using formula (3)

$⇒Im(Z₁\bar{Z₂}) = 0$

Using formula (1)

$⇒Im(Z₁\frac{|Z₂|²}{Z₂}) = 0$

$⇒|Z₂|²Im(\frac{Z₁}{Z₂}) = 0$

$⇒Im(\frac{Z₁}{Z₂}) = 0$