Equation of the ellipse is :
$\frac{x^2}{25}+\frac{y^2}{16}=1$
Equation of the chord having $\left(\frac{1}{2} \frac{3}{5}\right)$ as midpoint is:
$\begin{aligned}
& T=S_1 \\
& \Rightarrow \frac{\left(\frac{1}{4}\right)}{25}+\frac{\left(\frac{4}{25}\right)}{16}-1=\frac{\left(\frac{1}{2}\right) x}{25}+\frac{\left(\frac{2}{5}\right) y}{16}-1 \\
& \Rightarrow 4 x+5 y=4 \\
& \Rightarrow 5 y=4(1-x) \ldots(2)
\end{aligned}$
Solving with equation (1), we get
$\begin{aligned}
& 16 x^2+16(1-x)^2=400 \\
& \Rightarrow x^2-x-12=0 \\
& \Rightarrow x=4,-3
\end{aligned}$
From equation ( 2 ),
For $x=4, y=-\frac{12}{5}$
For $x=-3, y=\frac{16}{5}$
Therefore, length of the chord
$=\sqrt{7^2+\left(\frac{28}{5}\right)^2}=7 \sqrt{\frac{41}{25}} \text { unit }$
Hence, $\mathbf{k}=41$