If \(a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0\) represents a joint equation of directrices
of the hyperbola \(16 x^{2}-9 y^{2}=144\), then \(g+f-c=\)

Equation of the hyperbola: \(16 x^{2}-9 y^{2}=-144\)
This can be rewritten in the following way:
\(
\frac{x^{2}}{9}-\frac{y^{2}}{16}=-1
\)
This is the standard equation of a hyperbola, where
\(
\begin{array}{l}
a^{2}=9 \text { and } b^{2}=16 \\
\Rightarrow a^{2}=b^{2}\left(e^{2}-1\right) \\
\Rightarrow 9=16\left(e^{2}-1\right) \\
\Rightarrow e^{2}-1=\frac{9}{16} \\
\Rightarrow e^{2}=\frac{25}{16} \\
\Rightarrow e=\frac{5}{4}
\end{array}
\)
Coordinates of foci are given by \((0, \pm a e)\), i.e.
\((0, \pm 5)\).
Equation of the directrices: \(y=\pm \frac{a}{e}\)
\(\Rightarrow y=\pm \frac{4}{\frac{5}{4}}\) \(\Rightarrow 5 y \pm 16=0\)
Length of the latus rectum of the hyperbola \(=\frac{2 a^{2}}{b}\)
Length of the latus rectum \(=\frac{2 \times 9}{4}=\frac{9}{2}\).