Locus of the middle points of the chords of the hyperbola x^ 2 a^ 2...

$T = {S}_{1}$ at mid point $\left(h, k\right)\dfrac{hx}{a^{2}} - \dfrac{ky}{b^{2}} = \dfrac{h^{2}}{a^{2}} - \dfrac{k^{2}}{b^{2}}$

pass \left(\alpha, \beta\right)$point<br/><br/>$∴ \dfrac{h\alpha}{a^{2}} – \dfrac{ky}{b^{2}} = \dfrac{h^{2}}{a^{2}} – \dfrac{k^{2}}{b^{2}}$<br/><br/>$∴$locus is$\dfrac{x^{2} - \alphax}{a^{2}} - \dfrac{y^{2} - \betay}{b^{2}} = 0$<br/><br/>$⇒ \dfrac{{\left(x - \alpha / 2\right)}^{2}}{a^{2}} - \dfrac{{\left(y - \beta / 2\right)}^{2}}{b^{2}} = \dfrac{1}{4} \left(\dfrac{{\alpha}^{2}}{a^{2}} - \dfrac{{\beta}^{2}}{b^{2}}\right)

$= k^{2} \left(let\right)$

is a hyperbola.

Where centre is $\left(\alpha/ 2, \beta/ 2\right)$ .