$\mathrm{The} \mathrm{coordinates} \mathrm{of} \mathrm{a} \mathrm{point} \mathrm{common} \mathrm{to} \mathrm{a} \mathrm{directrix} \mathrm{and} \mathrm{an} \mathrm{asymptote} \mathrm{of} \mathrm{the} \mathrm{hyperbola} {\mathrm{x}}^2/25-{\mathrm{y}}^2/16=1 \mathrm{are}$

$$\begin{gathered} Given hyper bola \frac{x^2}{25}-\frac{y^2}{16}=1 where a^2=25,b^2=16 \\ Now e=\sqrt{\frac{41}{25}}=\frac{\sqrt{41}}{5} \\ Now equations of directrices are x=\pm \frac{a}{e}=\frac{\pm 25}{\sqrt{41}}\to \left(1\right) \\ Equation of asymptote is y=\pm \frac{b}{a}x\Rightarrow y=\pm \frac{4}{5}x\to \left(2\right) \\ Now point of intersection of \left(1\right) and \left(2\right) are \end{gathered}$$

$\left(\frac{25}{\sqrt{41}},\frac{20}{\sqrt{41}}\right)$ and $\left(\frac{-25}{\sqrt{41}},\frac{-20}{\sqrt{41}}\right)$