HyperbolaHardBloom L3
Question
The equation to the chord joining two points $\left( x_{1},y_{1} \right)$ and $\left( x_{2},y_{2} \right)$ on the rectangular hyperbola $xy = c^{2}$ is :
Options
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.$\frac{x}{y_{1} + y_{2}} + \frac{y}{y_{1} - y_{2}} = 1$
D.$\frac{x}{y_{1} - y_{2}} + \frac{y}{x_{1} - x_{2}} = 1$
Solution
Sol. (a) Equation of chord whose mid-point is $(h,k)$ is
$$\frac{xk + yh}{2} = c^{2} = hk$$
$$\frac{x}{h} + \frac{y}{k} = 2$$
Put $h = \frac{x_{1} + x_{2}}{2},k = \frac{y_{1} + y_{2}}{2}$
$$\Rightarrow \frac{x}{x_{1} + x_{2}} + \frac{y}{y_{1} + y_{2}} = 1$$
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