The length of the chord of the ellipse x ^ 2 25 + y ^ 2 16 = 1 ,...

Equation of chord with given middle point is given by,

$T = {S}_{1}$

$⇒ \dfrac{x}{25} + \dfrac{y}{40} = \dfrac{1}{25} + \dfrac{1}{100}$

$⇒ \dfrac{8x + 5y}{200} = \dfrac{8 + 2}{200}$

$⇒ y = \dfrac{10 - 8x}{5} . . \left(i\right)$

$⇒ \dfrac{{x}^{2}}{25} + \dfrac{( 10 - 8x{)}^{2}}{400} = 1$

$⇒ \dfrac{16{x}^{2} + 100 + 64{x}^{2} - 160x}{400} = 1$

$⇒ 4{x}^{2} - 8x - 15 = 0$

$⇒ x = \dfrac{8 \pm \sqrt{304}}{8}$

$⇒ {x}_{1} = \dfrac{8 + \sqrt{304}}{8}; {x}_{2} = \dfrac{8 - \sqrt{304}}{8}$

Similarly, $y = \dfrac{10 - 8 \pm \sqrt{304}}{5} = \dfrac{2 \pm \sqrt{304}}{5}$

$⇒ {y}_{1} = \dfrac{2 - \sqrt{304}}{5}; {y}_{2} = \dfrac{2 + \sqrt{304}}{5}$

Distance $= \sqrt{{\left({x}_{1} - {x}_{2}\right)}^{2} + {\left({y}_{1} - {y}_{2}\right)}^{2}}$

$= \sqrt{\dfrac{4 \times 304}{64} + \dfrac{4 \times 304}{25}}$

$= \dfrac{\sqrt{1691}}{5}$