If S and S’ are the foci of the ellipse $\frac{{\mathrm{x}}^2}{25}+\frac{{\mathrm{y}}^2}{16}=1,$ and P is any point on it then  range of values of $\mathrm{SP}.\mathrm{S}’\mathrm{P}$ is 

 
 

eq of an ellipse

$$\begin{gathered} \frac{{\mathrm{x}}^2}{25}+\frac{{\mathrm{y}}^2}{16}=1 \\ \mathrm{e}=\sqrt{\frac{25-16}{25}}=\frac{3}{5} \\ \mathrm{SP}=\left|\mathrm{a}-{\mathrm{ex}}_1\right| \\ =\left|\mathrm{a}-\mathrm{e}(\mathrm{a} \mathrm{cos\theta })\right| \\ =\left|5-3\mathrm{cos\theta }\right| \\ {\mathrm{S}}^1\mathrm{P}=\left|\mathrm{a}+{\mathrm{ex}}_1\right| \\ =\left|5+3\mathrm{cos\theta }\right| \\ \mathrm{SP}.{\mathrm{S}}^1\mathrm{P}=\left|25-9{\cos }^2\mathrm{\theta }\right| \\ {\cos }^2\mathrm{\theta } \min \mathrm{and} \max values are 0, 1 \\ 16\le \text{⨍ }\left(\mathrm{\theta }\right)\le 25 \end{gathered}$$