Algebra: Solving Problem on Complex Plane

Question

I have a math summer HW packet with algebra and precalculus but I'm really having trouble figuring out these two problems. If anyone could explain the method to do them I would appreciate it greatly.

1. A complex number $z$ is given by $z = \frac{a+i}{a-i}$, $a \in\mathbb{R}$.

(a) Determine the set of values of $a$ such that

(i) $z$ is real;

(ii) $z$ is purely imaginary.

(b) Show that $|z|$ is constant for all values of $a$.

2. Given that $z = \frac{2-i}{1+i} - \frac{6+8i}{u+i}$, find the values of $u \in \mathbb{R}$, such that $\Re(z) = \Im(z)$.

Answer 1

Hint, \begin{align} z &= \frac{a+i}{a-i} \\ &= \frac{(a+i)(a+i)}{(a+i)(a-i)} \\ &= \frac{(a+i)^2}{a^2+1} \\ &= \frac{a^2+2ai-1}{a^2+1} \end{align}

For this $z$ to be real, the coefficient of $i$ has to be....?

Answer 2

Hint:

(1)(a) Use complex conjugate so that you can write $z$ in Cartesian form.

(1)(b) Convert the Cartesian form you get in (a) to polar form

(2) Convert $z$ to polar form. Figure out why $\arg(z)$ being either $\pi/4$ or $5\pi/4$ will help you.

Answer 3

Let $a=r\cos t,1=r\sin t,\tan t=\dfrac1a$ Where $r>0,t$ are real

$$z=\dfrac{a+i}{a-i}=\cos2t+i\sin2t$$

$|z|=?$

When $z$ will be purely imaginary?