Area of the locus

Let $X=a+e^{i\alpha} \quad Y=b+3e^{i\beta}$

$M=\frac{X+Y}{2}=\frac{a+b}{2}+\frac{e^{i\alpha}+3e^{i\beta}}{2}$

Now let $\alpha$ and $\beta$ vary for $-\pi \leq \alpha , \beta \leq \pi\quad$ then $\frac{e^{i\alpha}+3e^{i\beta}}{2}$ is simply a circle of radius $\frac{1}{2}$ with its centre on the circumference of a circle with radius $\frac{3}{2}$ . So the locus of points is an annulus with inner radius $1$ and outer radius $2$ .

Area of the annulus is $\pi(2^2-1^2)=3\pi \approx \boxed{9.42}$

Discretizing the two angular parameters and plotting results in a cool looking pattern. In the first plot, the angle step is $\frac{2 \pi}{50}$ , and the plotting is done with two nested "for" loops; one for each parameter. In the second plot, the angle step is $\frac{2 \pi}{100}$ , and the curves fill the space between the inner and outer circles more thoroughly.

Of course, this "filling" is illusory, given that the locus itself is one-dimensional. Maybe we could define the "filling" (for a particular angle step size) by asking "What is the largest circle within the region bounded by the two greater circles, such that the locus does not pass through it?"