Question

The equation of the chord joining two points $ ({{x}_{1}},{{y}_{2}}) $ 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: (A)

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) $ $ \Rightarrow $ $ 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 $