Question

The equation of the chord joining two points (x₁, y₁) and (x₂, y₂) on the rectangular hyperbola xy = c² 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)

$\frac{x}{x_{1} + x_{2}} + \frac{y}{y_{1} + y_{2}} = 1$

Answer 2

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'

$x\left( \frac{y_{1} + y_{2}}{2} \right) + y\left( \frac{x_{1} + x_{2}}{2} \right) = 2\left( \frac{x_{1} + x_{2}}{2} \right)\left( \frac{y_{1} + y_{2}}{2} \right)$

⇒ x(y₁ + y₂) + y(x₁ + x₂) = (x₁ + x₂) (y₁ + y₂)

⇒ $\frac{x}{x_{1} + x_{2}} + \frac{y}{y_{1} + y_{2}} = 1$.