= Solution
The $xy$ block has characteristic <polynomial> $(2-\lambda)^2-a^2$. A convenient <orthonormal eigenbasis> for the full real <symmetric matrix> is
$$
e_+=\frac1{\sqrt2}(1,1,0)^T,\qquad e_-=\frac1{\sqrt2}(1,-1,0)^T,\qquad e_z=(0,0,1)^T,
$$
with <eigenvalues> $2-a$, $2+a$, and $c$, respectively. Hence its <orthogonal diagonalization> is
$$
\boxed{P^TAP=\operatorname{diag}(2-a,2+a,c),\qquad
P=\begin{pmatrix}1/\sqrt2&1/\sqrt2&0\\1/\sqrt2&-1/\sqrt2&0\\0&0&1\end{pmatrix}.}
$$
For repeated <eigenvalues>, every nonzero combination within the corresponding <eigenspace> is also an <eigenvector>; the displayed basis remains valid. Introduce <principal axes> coordinates
$$
X=\frac{x+y}{\sqrt2},\qquad Y=\frac{x-y}{\sqrt2},\qquad Z=z.
$$
The <quadratic form> becomes
$$
(2-a)X^2+(2+a)Y^2+cZ^2=1.
$$
The <principal-axis reduction of a quadric> therefore has three regimes.
For $0<a<2$, all coefficients are positive: the surface is an <ellipsoid> centred at the origin, with
$$
\boxed{\text{semiaxes }\frac1{\sqrt{2-a}},\quad\frac1{\sqrt{2+a}},\quad\frac1{\sqrt c}
\quad\text{along }e_+,e_-,e_z.}
$$
For $a=2$, the $X$ term disappears. The surface is an <elliptic cylinder>, unbounded along $e_+$, whose cross-section is $4Y^2+cZ^2=1$:
$$
\boxed{\text{cylinder axis }e_+,\qquad\text{cross-section semiaxes }\frac12,\ \frac1{\sqrt c}.}
$$
For $a>2$, there is one negative coefficient, so the surface is a one-sheet <hyperboloid>:
$$
\boxed{(a+2)Y^2+cZ^2-(a-2)X^2=1,\qquad\text{axis }e_+.}
$$
Its waist at $X=0$ is an ellipse with semiaxes $(a+2)^{-1/2}$ and $c^{-1/2}$. At other $X$, both waist radii are multiplied by $\sqrt{1+(a-2)X^2}$. There is no intercept on its longitudinal axis. Its asymptotic cone is $(a+2)Y^2+cZ^2=(a-2)X^2$. If positive <eigenvalues> coincide, the corresponding choice of <principal axes> is not unique, reflecting a rotational symmetry of the surface.
\Image[/past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2017/paper-1-q5-principal-quadrics.png]
{title=Ellipsoid, elliptic cylinder, and one-sheet hyperboloid in principal coordinates}
{description=Original calculated examples with $c=1$ and $a=1,2,3$. The horizontal longitudinal coordinate $X$ is directed along $(1,1,0)/\sqrt2$ in the original coordinates. Cylinder and hyperboloid plots show finite portions of unbounded surfaces.}
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