Question

The equation of the chord joining two points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ on the rectangular hyperbola $x y=c^{2}$ 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 $T=S_{1}$
or $x\left(\frac{y_{1}+y_{2}}{2}\right)+y\left(\frac{x_{1}+x_{2}}{2}\right)$
$=2\left(\frac{x_{1}+x_{2}}{2}\right)\left(\frac{y_{1}+y_{2}}{2}\right)$
$\Rightarrow x\left(y_{1}+y_{2}\right)+y\left(x_{1}+x_{2}\right)$
$=\left(x_{1}+x_{2}\right)\left(y_{1}+y_{2}\right)$
$\Rightarrow \frac{x}{x_{1}+x_{2}}+\frac{y}{y_{1}+y_{2}}=1$