Question

Which of the following is correct for a rectangular hyperbola
$xy=18$?

• A. Foci: $(6,6)\text{and}(-6,-6)$
• B. Eccentricity $=\sqrt{2}$
• C. Directrix: $x+y=\pm 6$
• D. Focal length$\sqrt{288}$

Answer 1

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

Focal length$\sqrt{288}$

Given, $xy=18\Rightarrow c=3\sqrt{2}$

Foci: $(\sqrt{2}c,\sqrt{2}c)$ and $(-\sqrt{2}c,-\sqrt{2}c)$

Foci:$(6,6)$ and $(-6,-6)$

$\text{Focal length}=4c\text{Focal length}=4\times 3\sqrt{2}\text{Focal length}=12\sqrt{2}\text{Focal length}=\sqrt{288}$

Directrix: $x+y=\pm \sqrt{2}c$
$\Rightarrow x+y=\pm \sqrt{2}\times 3\sqrt{2}$
$\Rightarrow x+y=\pm 6$

For a rectangular hyperbola
Eccentricity $=\sqrt{2}$