Complex Numbers. Prove that $z_1$ and $z_2$ lie on a straight line or not

Question

8. Let $z_1, z_2 \in \mathbb{C}\setminus\{0\}$. Suppose that $z_1\cdot z_2 \in \mathbb{R}$ Is it true that $z_1$ and $z_2$ lie on a straight line passing through $0$ in the Argand plane. If it is true, give a proof; otherwise give a counterexample.

Answer 1

The result is false.

Let $z_1 = 1+i$ and $z_2=1-i$. These two points lie on the straight line $x=1$ which does not pass through the origin.

Answer 2

$z_1\ne 0\land z_2\ne0\land z_1z_2\in\Bbb R\iff z_1\ne 0\land z_2\ne0\land\exists\alpha\in\Bbb R,\ z_1=\alpha \overline z_2$

So that will never be the case, unless $\overline z_2$ is a real multiple of $z_2$.

Answer 3

Remember that $Z_1*Z_2\in R$ then $Z_2$ will be in the direction of conjugate of $Z_1$ $\\$ Hence $Z_1\, , \,Z_2$ will be on a straight line passing through origin only if $Z_1$ is purely imaginary number.

Answer 4

$z_1z_2=r_1r_2e^{\theta_1 +\theta_2}$, where $z_1=r_1e^{i\theta_1}$ and $z_2=r_2e^{i\theta_2}$.

So $z_1z_2 \in\Bbb R\iff\theta_1 =-\theta_2 +k\pi$.

But, $z_1$ and $z_2$ lie on a line through the origin iff $\theta_1 =\theta_2 $.

Thus there are many counterexamples. For instance, let $\theta_1=-\theta_2\neq0 $.