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Q.
The equation of the chord joining two points$ (x_1, y_1) $ and $(x_2, y_2)$ on the rectangular hyperbola $xy = c^2$ is
Conic Sections
Solution:
The mid -point of the chord is $ \left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)$ The equation of the chord in terms if its mid-point is $T=S_{1}$,
i.e., $x\left(\frac{y_{1}+y_{2}}{2}\right)+y\left(\frac{x_{1}+x_{2}}{2}\right) $
$ = 2\left(\frac{x_{1}+x_{2}}{2}\right)\left(\frac{y_{1}+y_{2}}{2}\right)$
$ \Rightarrow x\left(y_{1}+y_{2}\right)+y\left(x_{1}+x_{2}\right)\left(y_{1}+y_{2}\right) $
$\Rightarrow \frac{x}{x_{1}+x_{2}}+\frac{y}{y_{1}+y_{2}} = 1$