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Q.
The locus of the mid points of the chords of the standard hypererbola passing through a fixed point ( $\alpha$, $\beta)$, is
Conic Sections
Solution:
Use $T = S _1$ which gives $\frac{ xh }{ a ^2}-\frac{ ky }{ b ^2}=\frac{ h ^2}{ a ^2}-\frac{ k ^2}{ b ^2}$ passes through $(\alpha, \beta)$
$\Rightarrow \frac{\alpha h}{a^2}-\frac{k \beta}{b^2}=\frac{h^2}{a^2}-\frac{k^2}{b^2} \Rightarrow \frac{\left(x-\frac{\alpha}{2}\right)^2}{a^2}-\frac{\left(y-\frac{\beta}{2}\right)^2}{b^2}=\frac{1}{4}\left(\frac{\alpha^2}{a^2}+\frac{\beta^2}{b^2}\right) \Rightarrow C$