1. Write the hyperbola in standard form

Given: $16x^2-9y^2=144$

Divide by $144$: $\frac{x^2}{9}-\frac{y^2}{16}=1$

So the hyperbola is of the form $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ with $a^2=9,\quad b^2=16 \Rightarrow a=3,\ b=4.$

2. Find the eccentricity

For a hyperbola, $c^2=a^2+b^2$ So, $c^2=9+16=25 \Rightarrow c=5.$

Also, $e=\frac{c}{a}=\frac{5}{3}.$

3. Use the directrix formula

For the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,$ the directrices are $x=\pm \frac{a}{e}.$

Now, $\frac{a}{e}=\frac{3}{5/3}=\frac{9}{5}.$

Hence the directrices are $x=\pm \frac{9}{5}.$

Given directrix: $5x+9=0 \Rightarrow x=-\frac{9}{5}$ This is the left directrix.

4. Find the corresponding focus

The foci of the hyperbola are $(\pm c,0)=(\pm 5,0).$

The left directrix $x=-\frac{9}{5}$ corresponds to the left focus: $(-5,0).$

5. Match with options

$(-5,0)$ is Option C.