MATH SOLVE

3 months ago

Q:
# The graph represents two complex numbers, z1 and z2. z1 = 0 β 4i and z2 = 2 β 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).

Accepted Solution

A:

Answer:[tex]\boxed{12/13 - (8/13)i;\quad 12/13, \; (8/13)i}[/tex]Step-by-step explanation:zβ = 0 - 4i
zβ = 2 - 3i
1. Quotient of zβ/zβ
[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 zβ/zβ
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]