= Solution
Along a curve $s\mapsto(x(s),y(s))$, the <chain rule> gives
$$
\frac d{ds}u_x=u_{xx}\dot x+u_{xy}\dot y,
\qquad
\frac d{ds}u_y=u_{xy}\dot x+u_{yy}\dot y.
$$
Together with the <second-order partial differential equation>, these form a <linear system> for $(u_{xx},u_{xy},u_{yy})$ whose coefficient matrix is
$$
\begin{pmatrix}
\dot x&\dot y&0\\
0&\dot x&\dot y\\
a&2b&c
\end{pmatrix}.
$$
A <characteristic curve> is precisely one along which this system is singular, equivalently where the <principal symbol> vanishes on a normal covector. Taking the <determinant> yields
$$
\boxed{a\dot y^2-2b\dot x\dot y+c\dot x^2=0.}
$$
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