Question

The equation of the directrices of the rectangular hyperbola $xy=c^2$ :

• A. $x+y=c\sqrt{2}$
• B. $x+y=c$
• C. $x-y=c\sqrt{2}$
• D. $x-y=c$

Answer 1

The correct option is A $x+y=c\sqrt{2}$

for rectangular hyperbola, $xy=c^2$
$\Rightarrow$$b=a=c\sqrt{2}$
Therefore, the co-ordinates of foci are given by $x=y=\pm \frac{a^2+a^2}{2a}$
$=\pm a=\pm c\sqrt{2}$
$\therefore$ Equation of direction for hyperbola $x^2-y^2=a^2$ is
$x$$=\pm \frac{a}{e}=\pm \frac{a}{\sqrt{2}}$
Now, rotating the axes through $-{45}^O$ that is writing $x\cos (-{45}^O)-y\sin (-{45}^O)$ for $x$.
$\therefore \frac{x+y}{\sqrt{2}}=\pm \frac{a}{\sqrt{2}}\Rightarrow x+y=\pm a$
$\Rightarrow x+y=\pm c\sqrt{2}$