If a rectangular hyperbola have the equation , $x y = c^{2}$ . Then find the locus of the middle points of the chords of constant length $2 d$ :

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Solution

Let mid point of chord of given hyperbola is $(h, k)$
Also let $(c t_{1}, \frac{c}{t_{1}}) a n d (c t_{1}, \frac{c}{t_{1}})$ be the end points of the chord.
then $2 h = (t_{1} + t_{2}) a n d 2 k = c (\frac{1}{t_{1}} + \frac{1}{t_{2}})$
According to equation
$c^{2} (t_{1} - t_{2})^{2} + c^{2} (\frac{1}{t_{1}} + \frac{1}{t_{2}}) = 4 d^{2}$
$\Rightarrow c^{2} [(t_{1} + t_{2})^{2} - 4 t_{1} t_{2}] [1 + \frac{1}{(t_{1} t_{2})^{2}}] = 4 d^{2}$
$\Rightarrow c^{2} [\frac{4 h^{2}}{c^{2}} - \frac{4 h}{k}] [1 + \frac{k^{2}}{h^{2}}] = 4 d^{2}$
$\Rightarrow (x y - c^{2}) (x^{2} + y^{2}) = d^{2} x y$

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