Locus of complex number.

Question

I have a locus of points $Z$ that satisfy equation: $Z = a + bt + ct^2$ where $a,b,c \in \mathbb{C}$, $t \in \mathbb{R}$ is a parameter and $\frac{b}{c} \in \mathbb{R}$, but I don't know to proceed and I need step by step.

Answer 1

In

$$Z=a+bt+ct^2$$

first drop the $a$ term that induces only a "final" translation. Thus write the issue under the form of plotting the set of all $z$ of the form :

$$z=bt+ct^2=c(t+ut^2)$$

because $b=cu$ with $u \in \mathbb{R}$. Now what does multiplication by $c$, considered under its polar form $c=re^{i \theta}$ ? Answer : a rotation and a homothety, that we can also do apart. It remains

$$s=t+ut^2=t(1+ut)$$

The locus of $s$ when $t \in (-\infty,+\infty)$ is either

• of the form $(x_0,+\infty)$ if $u>0$ (halfline of the real axis),

• of the form $(-\infty, x_0)$ if $u<$0 (halfline of the real axis),

• all the real axis is $u=0$.

In all cases this locus includes the origin (take $t=0$).

Thus proceeding in the reverse direction, we still get a halfline, somewhere in the so-called Argand plane. See picture below.

Fig. 1 : Case where the locus of $s$ is in green, the locus of $z$ is in red, and the (final) locus of $Z$ is in blue (with translational vector $a=4+4i$ in black). The enlargment factor (homothety) is $|c|=5.38$ ; the rotation value is $arg(c)=21.8°$.