= Solution
Choose the <kaon> momentum along $+x$ and the first <pion> along $+y$. <Momentum conservation> gives
$$
\mathbf p_1=(0,p_1),\qquad\mathbf p_2=(p,-p_1),
\qquad\boxed{p^2=p_2^2-p_1^2.}
$$
Both <pions> have mass $m_\pi$. Apply the on-shell condition to the second <pion>, using $E_2=E-E_1$:
$$
(E-E_1)^2-c^2(p^2+p_1^2)=m_\pi^2c^4.
$$
Substitute $E^2-p^2c^2=M^2c^4$ and $E_1^2-p_1^2c^2=m_\pi^2c^4$. The equal daughter masses cancel, leaving
$$
\boxed{E_1=\frac{M^2c^4}{2E},\qquad E_2=E-\frac{M^2c^4}{2E}.}
$$
With the supplied rest energies and $E=600$ MeV,
$$
\boxed{\begin{aligned}
pc&=335.253\ \mathrm{MeV},& E_1&=206.338\ \mathrm{MeV},\\
p_1c&=151.945\ \mathrm{MeV},& E_2&=393.662\ \mathrm{MeV},\\
p_2c&=368.078\ \mathrm{MeV}.&&
\end{aligned}}
$$
The second <pion> travels below the parent axis, at
$$
\boxed{\theta_2=-\tan^{-1}(p_1/p)=-24.381^\circ.}
$$
Reflection in the $x$ axis gives the equivalent opposite-sign configuration. The condition $E_1\geq m_\pi c^2$ requires $E\leq M^2c^2/(2m_\pi)\simeq886.84$ MeV for this perpendicular first-pion geometry. The given $600$ MeV lies within that range; the geometry cannot be extrapolated to an arbitrarily boosted <kaon>.
\Image[/past-exam-of-the-symmetry-and-relativity-course-of-the-university-of-oxford/2015-kaon-pion-decay-events-and-simultaneity.png]
{title=Pion decay tracks, a spacelike event separation and its simultaneous frame}
{description=Original calculation from the supplied masses and proper lifetimes. The laboratory separation is shown in its actual decay plane; a boost along that separation makes the two decay events simultaneous.}
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