Question

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:

• A. $ \frac{x}{{{x}_{1}}+{{x}_{2}}}+\frac{y}{{{y}_{1}}+{{y}_{2}}}=1 $
• B. $ \frac{x}{{{x}_{1}}-{{x}_{2}}}+\frac{y}{{{y}_{1}}-{{y}_{2}}}=1 $
• C. $ \frac{x}{{{y}_{1}}+{{y}_{2}}}+\frac{y}{{{x}_{1}}+{{x}_{2}}}=1 $
• D. $ \frac{x}{{{y}_{1}}-{{y}_{2}}}+\frac{y}{{{x}_{1}}-{{x}_{2}}}=1 $

Answer 1

Answer: (C)

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 of its mid-point is $ T=S $ . or $ 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) $
$ x({{y}_{1}}+{{y}_{2}})+y({{x}_{1}}+{{x}_{2}})=({{x}_{1}}+{{x}_{2}})({{y}_{1}}+{{y}_{2}}) $
$ \frac{x}{{{x}_{1}}+{{x}_{2}}}+\frac{y}{{{y}_{1}}+{{y}_{2}}}=1 $