How can I describe the set:
$$ \left\vert z - {\rm i}\,\right\vert = 3\left\vert z\right\vert $$
It does appear quite unfamiliar.
Attempt: $$ \left\vert\frac{z-i}{z}\right\vert = 3 $$ so, $$ \left\vert 1 - {\rm i}\,\frac{1}{z}\right\vert = 3 $$
But this seems to be even more difficult to visualize.
Hint: Square both sides, and write $z = x + iy$:
$$9(x^2 + y^2) = |x + i (y - 1)|^2 = x^2 + (y - 1)^2$$
Rearranging,
$$8x^2 + 8y^2 + 2y - 1 = 0$$
Divide through by $8$ and find
$$x^2 + y^2 + \frac{1}{4} y = \frac{1}{8}$$
or even better,
$$x^2 + y^2 + \frac{1}{4} y + \frac{1}{8} = \frac{1}{4}$$
Try rewriting the left side as the equation of a circle.
Observe the notion of distance when you write $|z-i|=a$ or $3|z|=b$.
The first equation describes a circle with radius $a$ and center on the point $(0,1)$. The second one describes a circle with radius $b/3$ and center at the origin.
$|z-i|=3|z|$ should describe a curve such that each point has it's distance to $(0,1)$ tree times the distance to the origin.
It will be a circle. More about this subject: http://en.wikipedia.org/wiki/Circles_of_Apollonius
