The product of lengths of perpendicular from any point on the hyperbola ${\mathrm{x}}^2 – {\mathrm{y}}^2 = 16$ to its asymptotes is

$$\begin{gathered} \mathrm{Given} \mathrm{hyperbola} {\mathrm{x}}^2-{\mathrm{y}}^2=16 \\ where a^2=b^2=16 \\ Now \mathrm{product} \mathrm{of} \mathrm{perpendiculars} \mathrm{from} \mathrm{any} \mathrm{point} \mathrm{on} \mathrm{hyperbola} \mathrm{to} \mathrm{its} \mathrm{asymptotes} \\ =\frac{{\mathrm{a}}^2{\mathrm{b}}^2}{{\mathrm{a}}^2+{\mathrm{b}}^2} \\ =\frac{16\times 16}{32} \\ =8 \end{gathered}$$