= Solution
{c}
Use the convention that positive vortex strength is counterclockwise and complex velocity is $u-iv$. A <complex potential of a two-dimensional potential flow> for the superposed <uniform flow> and two <point vortices> is
$$
\boxed{W(z)=Uz+\frac{\Gamma}{2\pi i}\log(z-a)-\frac{\Gamma}{2\pi i}\log(z+a).}
$$
Local choices of logarithm branch change no velocity. Differentiation gives
$$
W'(z)=U-\frac{i\Gamma a}{\pi(z^2-a^2)}.
$$
For nonzero vortex strength and $U>0$, its two zeros are
$$
\boxed{z_\pm=\pm\sqrt{a^2+\frac{i\Gamma a}{\pi U}}.}
$$
The number under the square root is nonzero, so these are distinct <stagnation points>, neither at a vortex.
A free <point vortex> moves with the velocity induced by the other vortex and the background, omitting its own singular field. At either vortex, the other induces vertical velocity $-\Gamma/(4\pi a)$, while the background contributes $(U,0)$. Their velocities are therefore equal. Their separation stays $(2a,0)$, making this induced velocity constant for all time. Both travel along parallel straight lines with
$$
\boxed{\mathbf V=\left(U,-\frac{\Gamma}{4\pi a}\right),\qquad
|\mathbf V|=\sqrt{U^2+\frac{\Gamma^2}{16\pi^2a^2}}.}
$$
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