Question

For the hyperbola $xy=4$, which of the following option(s) is(are) true

• A. Equation of transverse axis is $y\pm x=0$
• B. Eccentricity $e=\sqrt{2}$
• C. Coordinates of foci are $(\pm 2\sqrt{2},\pm 2\sqrt{2})$ and equation of the directrix is given by $x+y\pm 2\sqrt{2}=0$
• D. Length of latus rectum $=4\sqrt{2}$

Answer 1

Answer: (B), (C), (D)

Eccentricity $e=\sqrt{2}$
Coordinates of foci are $(\pm 2\sqrt{2},\pm 2\sqrt{2})$ and equation of the directrix is given by $x+y\pm 2\sqrt{2}=0$
Length of latus rectum $=4\sqrt{2}$

For a rectangular hyperbola of the form $xy=c^2$ , the equation of the traverse axis is $y=x$.
Hence, option A is incorrect.
The eccentricity is $\sqrt{2}$. Hence, option B is correct.
The coordinates of foci are $(c\sqrt{2},c\sqrt{2})$ . For this hyperbola, $c=2$. Hence, the coordinates of foci are $(2\sqrt{2},2\sqrt{2})$. Hence, option C is correct.
The length of the latus rectum is given by $2c\sqrt{2}$. Hence, the length of the latus rectum $4\sqrt{2}$. Hence, option D is also correct.