Q. 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) $\frac{x}{y_{1} + y_{2}} + \frac{y}{x_{1} + x_{2}} = 1$
Explanation
(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
$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)\$
$\left\lbrack \therefore T = S_{1} \right\rbrack$
$\Rightarrow \ \ \frac{x}{{x_{1} + x}_{2}} + \frac{y}{y_{1 + y_{2}}} = 1$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.