Ray emanating from the point (0,6) is incident on the ellipse $25{\mathrm{x}}^2+16{\mathrm{y}}^2=1600$ at the point P with ordinate 5. After reflection, ray cuts the y-axis at B. The length of PB is

$$\begin{gathered} \frac{{\mathrm{x}}^2}{64}+\frac{{\mathrm{y}}^2}{100}=1(\mathrm{a}<\mathrm{b}) \\ \Rightarrow e=\sqrt{\frac{100-64}{100}}=\sqrt{\frac{36}{100}} \end{gathered}$$
$$\begin{gathered} \text{ Focii }=\left(0,\pm 10,\frac{6}{10}\right)\text{ =(0,±6)} \\ \text{Let co-ordinates of }\mathrm{P}\text{ are }(\mathrm{\lambda },5) \end{gathered}$$
$$\begin{gathered} \frac{{\mathrm{\lambda }}^2}{64}+\frac{25}{100}=1 \\ \Rightarrow \mathrm{\lambda }=\pm 4\sqrt{3} \\ \therefore \mathrm{P}=(\pm 4\sqrt{3},5) by the reflexion property reflected ray passes through other focus (0,-6) \end{gathered}$$
$\therefore \mathrm{PB}=\sqrt{(4\sqrt{3}{)}^2+(5+6{)}^2}=13.$