Algebra
# Complex Numbers

The expression \( \displaystyle \frac{1}{2-11i} \) can be rewritten with a real denominator (using the complex conjugate) in the form \[ \displaystyle \frac{a+bi}{c}, \]

where \( a \), \( b \), and \( c \) are positive, co-prime integers. What is the sum of \( a \), \( b \), and \( c \)?

**Details and assumptions**

\( i \) is the the imaginary unit, defined by \( i^2 = -1 \).

Consider the complex numbers \[A=12+i \mbox{ and } B=11-3i.\] If the value of \(2A\overline{A}+2\overline{A}B+A\overline{B}+B\overline{B}\) is \(X+Yi\), what is the value of \(X-Y\)?

**Details and assumptions**

\(\overline{A}\) denotes the complex conjugate of the complex number \(A\).

\(i\) is the imaginary number that satisfies \(i^2 = -1\).

For complex numbers \(\alpha=5+7i\) and \(\beta=3-i,\) what is the value of \[\alpha \overline{\alpha} +\overline{\alpha} \beta +\alpha \overline{\beta} +\beta \overline{\beta}?\]

**Details and assumptions**

\(\overline{\alpha}\) denotes the complex conjugate of the complex number \(\alpha\).

\(i\) is the imaginary number that satisfies \(i^2 = -1\).

Consider the complex numbers \[A=8-2i \mbox{ and } B=4-i.\] What is the value of \((A+B)^2+\left(\overline{A+B}\right)^2\)?

**Details and assumptions**

\(\overline{A}\) denotes the complex conjugate of the complex number \(A\).

\(i\) is the imaginary number that satisfies \(i^2 = -1\).

Let \(\omega\) and \(z\) be the complex numbers \[ \omega =2-i, \quad z =\frac{\omega+4}{4\omega-1}. \]

If \(z\overline{z}\) can be expressed as \(\frac{a}{b},\) where \(a\) and \(b\) are coprime positive integers, what is \(a+b?\)

**Details and assumptions**

\(\overline{z}\) denotes the complex conjugate of the complex number \(z\).

\(i\) is the imaginary number that satisfies \(i^2 = -1\).

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