Consider an axis-parallel ellipsoid $E_d$ in ${\mathbb R}^d$ that is defined by an equation of the form $a_1x_1^2+\cdots+a_dx_d^2 = r^2$. Consider a hyperplane $h$ that is defined by the equation $b_1x_1+\cdots + b_d x_d = b'$. Assume that $a_1,\ldots,a_d,r^2,b_1,\ldots,b_d,b'$ all rational numbers. Finally, set $E_{d-1} = E_d\cap h_d$.
Given that $E_{d-1}$ is a $(d-2)$-dimensional ellipsoid (as expected), we consider it as living in ${\mathbb R}^{d-1}$ (that is, in $h_d$). Is it true that $E_{d-1}$ can desribed using an equation with rational coefficients? (Moving to ${\mathbb R}^{d-1}$ makes $E_{d-1}$ a hypersurface so it is again defined by a single equation. We can also rotate the space to have $E_{d-1}$ axis parallel.)