Explanation

(c)
Equation of an ellipse is
${16x}^{2} + {25y}^{2} = 400$
$\Rightarrow \frac{x^{2}}{25} + \frac{y^{2}}{16} = 1$
Here, $a^{2} = 25$ and $b^{2} = 16$
But $b^{2} = a^{2}(1 - e^{2})$
$\Rightarrow 16 = 25\left( 1 - e^{2} \right) \Rightarrow \frac{16}{25} = 1 - e^{2}$
$\Rightarrow e^{2} = \frac{9}{25} \Rightarrow e = \frac{3}{5}$
Now, foci of the ellipse are $(3,\ 0)$
Now, ${PF}_{1} + {PF}_{2} = 2a = 2 \times 5 = 10$