Given ellipse

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

Compare with standard form:

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

so

$a^2=25,\quad b^2=16.$

Midpoint of the chord

The midpoint of the required chord is

$M\left(1,\frac{2}{5}\right).$

Equation of chord with given midpoint

For the ellipse

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$

the chord whose midpoint is $(x_1,y_1)$ is given by the midpoint form:

$\frac{xx_1}{a^2}+\frac{yy_1}{b^2}=\frac{x_1^2}{a^2}+\frac{y_1^2}{b^2}.$

Substitute $x_1=1$, $y_1=\frac25$, $a^2=25$, $b^2=16$:

$\frac{x}{25}+\frac{y(2/5)}{16}=\frac{1}{25}+\frac{(2/5)^2}{16}.$

Simplify:

$\frac{x}{25}+\frac{y}{40}=\frac{1}{25}+\frac{4}{25\cdot 16}$ $\frac{x}{25}+\frac{y}{40}=\frac{1}{25}+\frac{1}{100}$ $\frac{x}{25}+\frac{y}{40}=\frac{1}{20}.$

Multiply by $200$:

$8x+5y=10.$

So the required chord lies on the line

$8x+5y-10=0.$

Find points of intersection with ellipse

From the line,

$y=2-\frac{8x}{5}.$

Substitute into ellipse:

$\frac{x^2}{25}+\frac{\left(2-\frac{8x}{5}\right)^2}{16}=1.$

Now,

$\left(2-\frac{8x}{5}\right)^2=4-\frac{32x}{5}+\frac{64x^2}{25}.$

Hence,

$\frac{x^2}{25}+\frac{1}{16}\left(4-\frac{32x}{5}+\frac{64x^2}{25}\right)=1.$

$\frac{x^2}{25}+\frac14-\frac{2x}{5}+\frac{4x^2}{25}=1.$

$\frac{5x^2}{25}-\frac{2x}{5}+\frac14=1$

$\frac{x^2}{5}-\frac{2x}{5}-\frac34=0.$

Multiply by $20$:

$4x^2-8x-15=0.$

Solve:

$x=\frac{8\pm\sqrt{64+240}}{8}=\frac{8\pm\sqrt{304}}{8}=1\pm\frac{\sqrt{19}}{2}.$

So

$x_1=1+\frac{\sqrt{19}}{2},\qquad x_2=1-\frac{\sqrt{19}}{2}.$

Therefore,

$\Delta x=x_1-x_2=\sqrt{19}.$

Find corresponding difference in $y$

Since

$y=2-\frac{8x}{5},$

the slope is $-\frac85$. Thus,

$\Delta y=-\frac85\,\Delta x=-\frac85\sqrt{19}.$

Hence,

$|\Delta y|=\frac85\sqrt{19}.$

Length of the chord

$L=\sqrt{(\Delta x)^2+(\Delta y)^2}$

$=\sqrt{19+\left(\frac85\sqrt{19}\right)^2}$

$=\sqrt{19+\frac{64}{25}\cdot 19}$

$=\sqrt{19\left(1+\frac{64}{25}\right)}$

$=\sqrt{19\cdot\frac{89}{25}}$

$=\frac{\sqrt{1691}}{5}.$

Compare with options

This matches:

$\boxed{\frac{\sqrt{1691}}{5}}$

So the correct option is A.