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The graph represents two complex numbers, z1 and z2.
z1 = 0 – 4i and z2 = 2 – 3i.
What are the real and imaginary parts of the conjugate of the quotient of z_1/z_2 ? Use / for the fraction bar(s).

The graph represents two complex numbers z1 and z2 z1 0 4i and z2 2 3i What are the real and imaginary parts of the conjugate of the quotient of z1z2 Use for th class=

Respuesta :

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Answer:

[tex]\boxed{12/13 - (8/13)i;\quad 12/13, \; (8/13)i}[/tex]

Step-by-step explanation:

z₁ = 0 - 4i

z₂ = 2 - 3i

1. Quotient of z₁/z₂

[tex]\begin{array}{lrcl}z_{1}/z_{2} & = & (0 - 4i)/(2 - 3i) &\\& =& -(4i)/(2 - 3i) \times (2 + 3i)/(2 + 3i)& \text{Multiply by conjugate}\\& =& -[4i(2 + 3i)]/[(2 - 3i)(2+ 3i)] &\text{Multply fractions} \\& =& -(8i - 12)/(4 + 9) &\text{Distribute and FOIL}\\& =& (12 - 8i)/13 &\text{Distribute and add}\\& =& \mathbf{12/13 - (8/13)i} & \text{Put into standard form}\\\end{array}\\\text{The quotient is }\boxed{\mathbf{12/13 - (8/13)i}}[/tex]

2. Conjugate of quotient of z₁/z₂

Change the sign of the imaginary part. The conjugate of the quotient becomes

[tex]12/13 + (8/13)i\\\text{The real part is $\boxed{\mathbf{12/13}}$ and the imaginary part is $\boxed{\mathbf{(8/13)i}}$}\\[/tex]

The needed conjugate of quotient as:

[tex]\overline{z} = \dfrac{12}{13} + \dfrac{8}{13}i[/tex]

The given complex numbers are:

[tex]z_1 = 0 - 4i\\z_2 = 2 - 3i[/tex]

Their quotient is calculated as:

[tex]\dfrac{z_1}{z_2} = \dfrac{0 - 4i}{2 - 3i}\\\\= \dfrac{(0-4i)(2+3i)}{(2-3i)(2+3i)}\\\\= \dfrac{(-8i-12i^2)}{(4 - 9i^2)}\\\\= \dfrac{12 - 8i}{4+9}\\\\= \dfrac{12}{13} -\dfrac{8}{13}i\\\\= z ( say)[/tex]

We used rationalization to simplify the quotient. It is done by multiplying and dividing by same factor so as to simplify the given fraction.

It is the property of iota that [tex]i^2 = -1[/tex]

Now conjugate of z is

[tex]\overline{z} = \dfrac{12}{13} + \dfrac{8}{13}i[/tex]

Thus we have needed conjugate of quotient as:

[tex]\overline{z} = \dfrac{12}{13} + \dfrac{8}{13}i[/tex]

Learn more here:

https://brainly.com/question/10468518

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