How to find the set of lines of fixed length connecting two circles? Suppose I have two circles as shown below.

I want to find the set of line segments of size $\mathbf{w}$ connecting the two circles.
What's a nice way to do this?
I thought of a simple, ugly way.  I could write out the equations
\begin{align}
(u_x - a)^2 + (u_y - b)^2 &= \lVert \mathbf{u}\rVert^2\\
(v_x - c)^2 + (v_y - d)^2 &= \lVert \mathbf{v}\rVert^2\\
\sqrt{(v_x - u_x)^2 + (v_y-u_y)^2}  &=\lVert \mathbf{w}\rVert 
\end{align}
The first two equations are the equations for the two circles, while the last formula is the distance formula for the line segment.
However, when I try to do the math, it's really ugly, so I was wondering if there's a nicer way to do this.
Geometrically, I think this basically amounts to treating the third equation like a circle with an origin centered on the edge of the first circle and then pivoting the third circle's origin around the edge of the first circle to trace out the line between them, like this:

I feel like there also might be a way to do this with matrices, but I'm not sure how I would do that.
$$\begin{bmatrix}
1 & 1 & -2a & -2b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 1 & -2c & -2d & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & -2 & 0 & 0 & -2 \\
\end{bmatrix} 
\begin{bmatrix}
u_x^2 \\
u_y^2 \\ 
u_x\\
u_y\\
v_x^2\\
v_y^2\\
v_x\\
v_y\\
u_xv_x\\
u_xv_y\\
u_yv_x\\
u_yv_u\\
\end{bmatrix}
=
\begin{bmatrix}
\lVert \mathbf{u}\rVert^2 - a^2 - b^2 \\
\lVert \mathbf{v}\rVert^2  - c^2 - d^2\\
\lVert \mathbf{w}\rVert^2
\end{bmatrix}
$$
But I'm not sure where I should go from there.

To depict full rotations we should express longer crank rotation as a function of the shorter. For constant $len$ of connecting rod (CR) we should have
$$(a \cos (t)-b \cos (u)+2 c)^2+(a \sin (t)-b \sin (u))^2=\text{len}^2 $$
$ u(t) $  is solved and plotted with help from Mathematica website.
The crank $a$ at right rotates all around but the longer crank $b$ at left oscillates  between two limits of angle $u$.
CR and crank lengths in the 4-Bar mechanism used:
$$len= w=7.5,2c=10, a= 2, b=3$$ Blue $u(t)$ in forward motion and red in backward of oscillation. Inclination of CR can be also found if needed... for this "Pumpjack Donkey".

