The equation to the hyperbola with vertex (5, 0) and having asymptotes $3x \pm 5y =0$ is

let hyperbola be $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}-\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$

Given Vertex $(\mathrm{a}, 0)=(5, 0) \Rightarrow \mathrm{a}=5$

Given asymptotes are $\mathrm{y}=\pm \frac{3}{5}\mathrm{x}$

compare with $\mathrm{y}=\pm \frac{\mathrm{b}}{\mathrm{a}}\mathrm{x}$

$$\begin{gathered} \Rightarrow \frac{\mathrm{b}}{\mathrm{a}}=\frac{3}{5} \\ \Rightarrow \frac{\mathrm{b}}{5}=\frac{3}{5} \\ \Rightarrow \mathrm{b}=3 \end{gathered}$$

$$\begin{gathered} \therefore \mathrm{hyperbola} \mathrm{is} \frac{{\mathrm{x}}^2}{25}-\frac{{\mathrm{y}}^2}{9}=1 \\ \Rightarrow 9{\mathrm{x}}^2-25{\mathrm{y}}^2=225 \end{gathered}$$