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

Here $a=5, b=4$ and the centre of the ellipse is $(0,0)$.

The major axis of ellipse is along the  $X-$axis and minor axis is along $Y-$axis.

The distance of point $P(x,y)$ from the centre of the ellipse $C$ is maximum, only when the point $P(x,y)$ is on the endpoint of major axis (or at the vertex of ellipse along the major-axis).

Therefore, coordinate of point $P$ at the endpoint of major-axis is $(5,0)$ or $(-5,0)$.

Hence, maximum value of $CP$ is $5$.

Similarly, the distance of point $P(x,y)$ from the centre of the ellipse is minimum, only when the point $P(x,y)$ is on the endpoint of minor axis (or at the vertex of ellipse along the minor-axis).

Therefore, the coordinate of point $P$ at the endpoint of minor-axis is $(4,0)$ or $(-4,0)$.

Hence, minimum value of $CP$ is $4$.

Therefore, the sum of maximum and minimum values of CP $=a+b$

$=5+4$

$= 9$.