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Question

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


A
x+y=c2
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B
x+y=c
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C
xy=c2
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D
xy=c
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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 $$x$$.
$$\displaystyle\therefore \frac { x+y }{ \sqrt { 2 }  } =\pm \frac { a }{ \sqrt{2}} \Rightarrow x+y=\pm a$$
$$\Rightarrow x+y=\pm c\sqrt { 2 } $$

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