Question

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

A
x+y=c2
B
x+y=c
C
xy=c2
D
xy=c

Solution

## 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$$x$$$$\displaystyle=\pm \frac { a }{ e } =\pm \frac { a }{ \sqrt { 2 } }$$Now, rotating the axes through $$-{ 45 }^{ O }$$ that is writing $$x\cos { \left( -{ 45 }^{ O } \right) } -y\sin { \left( -{ 45 }^{ O } \right) }$$ for $$x$$.$$\displaystyle\therefore \frac { x+y }{ \sqrt { 2 } } =\pm \frac { a }{ \sqrt{2}} \Rightarrow x+y=\pm a$$$$\Rightarrow x+y=\pm c\sqrt { 2 }$$Maths

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