The locus of the midpoint of the chords of the hyperbola (x2)/(25)...

The equation of the chord whose mid-point is $( h, k )$ is $T = {S}_{1}$
$⇒ \dfrac{xh}{25} - \dfrac{yk}{36} = \dfrac{h2}{25} - \dfrac{k2}{36}$
Since it passes through \left(2, 4\right)$<br />$\dfrac{2h}{25} - \dfrac{4k}{36} = \dfrac{h2}{25} - \dfrac{k2}{36} ⇒ \dfrac{h^{2} - 2h + 1}{25} - \dfrac{k^{2} - 4k + 4}{36} = \dfrac{1}{25} - \dfrac{4}{36}$<br />Hence, the locus of \left(h, k\right)$ is
$\dfrac{\left(x - 1\right)2}{\dfrac{16}{9}} - \dfrac{\left(y - 2\right)2}{\dfrac{64}{25}} = - 1$
This is the equation of a hyperbola whose transverse axis length is $\dfrac{16}{5}$& units