= Solution
Take $a>0$ as a length. For the rear hyperbola,
$$
f(t)=\sqrt{a^2+c^2t^2},\qquad
v(t)=\frac{c^2t}{f(t)},\qquad\gamma(t)=\frac{f(t)}a,
\qquad\boxed{\gamma v=\frac{c^2t}{a}.}
$$
The implicit front event from <4/b/iii> has $t_B=(1+L/a)t$ and $x_B=(1+L/a)f(t)$. Eliminate the rear time $t$ to obtain
$$
\boxed{h(t_B)=\sqrt{(a+L)^2+c^2t_B^2}.}
$$
The rear <proper acceleration> is $c^2/a$, while the front <proper acceleration> is $c^2/(a+L)$. Thus <Born rigidity> in the acceleration requires the front rocket to accelerate less strongly. Their constant separation $L$ is measured on the tilted instantaneous-rest slices; their equal-laboratory-time coordinate separation decreases from $L$ rather than staying constant. At $t=0$ both rockets are at rest, at $x=a$ and $x=a+L$.
\Image[/past-exam-of-the-symmetry-and-relativity-course-of-the-university-of-oxford/2015-born-rigid-rockets-and-light-frequency-shift.png]
{title=Born-rigid rocket hyperbolae, a received light pulse and the frequency shift}
{description=Original Minkowski diagram and analytic reception quantities. The illustrative geometry uses L over a equal to 0.8; the solution formulas apply to arbitrary positive a and L. The sloping dashed segment joins simultaneous comoving events.}
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