Conjugates

Conjugate product of complex numbers ●○○○○ 10 ptsThe complex numbers $a$ and $b$ satisfy $a \overline{b} = -1 + 5i$ . Find the real part and the imaginary part of $\overline{a}\, b$ .
Product of a complex number and its conjugate ●○○○○ 10 ptsGiven a complex number $z$ satisfying $z(1+i)=2i$ , find $z\cdot\overline{z}$ .
Product of a complex root and its conjugate ●○○○○ 10 ptsLet $z$ be a complex number that satisfies $z^2 + 3 = 0$ . Find the value of $z \cdot \overline{z}$ , where $\overline{z}$ denotes the complex conjugate of $z$ .
Minimise |z|² under a conjugate constraint ●●○○○ 20 ptsOver complex numbers $z$ , minimize $z^*z$ subject to $z = z^* + i$ , where $z^*$ is the complex conjugate of $z$ . Write the minimizing $z$ as $z = a + bi$ with real $a$ and $b$ , and find $a$ and $b$ .
Complex solutions of a conjugate equation ●●○○○ 20 ptsFind all complex numbers $z$ satisfying $z\bar z+z^2=\overline{\bar z}-z+2$ . List the solutions in increasing order.
Complex equation with a conjugate ●●○○○ 20 ptsLet $z$ be a complex number and let $z'$ denote its complex conjugate. Find $z$ if $\frac{z}{2}+3=\frac{z'}{3}-2$ .

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