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

The correct option is A $\frac{x}{x_1+x_2}+\frac{y}{y_1+y_2}=1$

The midpoint of the chord is $(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$
The equation of the chord in terms of its midpoint is
$T=S_1$
$\Rightarrow \frac{x}{2}\left(\frac{y_1+y_2}{2}\right)+\frac{y}{2}\left(\frac{x_1+x_2}{2}\right)-c^2=\left(\frac{x_1+x_2}{2}\right)\left(\frac{y_1+y_2}{2}\right)-c^2$
$\Rightarrow x(y_1+y_2)+y(x_1+x_2)=(x_1+x_2)(y_1+y_2)$
$\Rightarrow \frac{x}{x_1+x_2}+\frac{y}{y_1+y_2}=1$