= Solution
With the chosen basis, let $\mathbf b_1=(u,v)$. The <reciprocal lattice> conditions give
$$
a(u+v)=2\pi,\qquad a(-u+v)=0,
$$
so $u=v=\pi/a$. Similarly, $\mathbf a_1\cdot\mathbf b_2=0$ and $\mathbf a_2\cdot\mathbf b_2=2\pi$ give
$$
\boxed{\mathbf b_1=\frac{\pi}{a}(1,1),\qquad
\mathbf b_2=\frac{\pi}{a}(-1,1)}.
$$
All reciprocal points are
$$
\mathbf G=m\mathbf b_1+n\mathbf b_2
=\frac{\pi}{a}(m-n,m+n),\qquad m,n\in\mathbb Z.
$$
Thus, in coordinates measured in $\pi/a$, the two integer coordinates have the same parity. The <reciprocal lattice> is another rotated <square Bravais lattice>, with nearest-neighbor spacing $\sqrt2\pi/a$.
\Image[/past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2023/paper-2-reciprocal-zone.png]
{title=Reciprocal lattice, first zone and the inscribed Fermi circle}
{description=Original construction in coordinates $k_xa/\pi$ and $k_ya/\pi$. The reciprocal points have integer coordinates of the same parity. The shaded diamond is the origin-centered Wigner-Seitz cell; the circle has radius $1/\sqrt2$ in these units and touches its four sides.}
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