1. Introduction and Key Concept: Equation of a Chord with a Given Midpoint

The problem asks us to determine the length of a chord of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$, given that its midpoint is $\left(1, \frac{2}{5}\right)$. A fundamental and highly efficient approach for problems involving chords and their midpoints in conic sections is to utilize the $T=S_1$ formula.

For any general conic section represented by the equation $S = Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, the equation of a chord whose midpoint is $(x_1, y_1)$ is given by the formula: $T = S_1$ Let's define $T$ and $S_1$ specifically for an ellipse in the standard form $\frac{x^2}{a^2} + \frac{y^2}{b^2