If a point P(x, y) moves along the ellipse $\frac{{\mathrm{x}}^2}{25}+\frac{{\mathrm{y}}^2}{16}=1$ and if C is the center of the ellipse, then the sum of maximum and minimum values of CP is

$Given ellipse \frac{x^2}{25}+\frac{y^2}{16}=1$

Let $\mathrm{P}(5\mathrm{cos\theta }, 4\mathrm{sin\theta })$ lies on ellipse

$$\begin{gathered} \mathrm{centre} C=(0, 0) \\ C{P}^2=25{\cos }^2\mathrm{\theta }+16{\sin }^2\mathrm{\theta } \\ =9{\cos }^2\mathrm{\theta }+16 \\ \Rightarrow CP=\sqrt{9{\cos }^2\mathrm{\theta }+16} \\ \min imum and \max imum \mathrm{values} \mathrm{of} {\cos }^2\mathrm{\theta } \mathrm{is} 0, 1 \\ \min , \max \mathrm{values} \mathrm{of} CP \mathrm{is} \sqrt{0+16}, \sqrt{9+16}=4, 5 \\ \mathrm{sum}=4+5=9 \end{gathered}$$