Question

For the hyperbola $xy=-16$. Which of the following is/are true $?$

• A. Length of latus rectum is $8\sqrt{2}$ unit
• B. coordinates of vertices will be $(-4,4)$ and $(4,-4)$
• C. equation of directrices will be $x+y=\pm 4\sqrt{2}$
• D. Length of transverse axis is $8\sqrt{2}$ unit

Answer 1

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

The correct options are
A Length of latus rectum is $8\sqrt{2}$ unit
B coordinates of vertices will be $(-4,4)$ and $(4,-4)$
D Length of transverse axis is $8\sqrt{2}$ unit

$xy=-16$
$\Rightarrow c=4$
coordinates of vertices will be points of intersection of transverse axis $x+y=0$ and $xy=-16$ i.e.,
$(-4,4)$ and $(4,-4)$
Length of transverse axis is
$2\sqrt{2}c=8\sqrt{2}$ unit
Now, equation of directrices will be $x-y=\pm c\sqrt{2}=\pm 4\sqrt{2}$
Length of latus rectum $=$Length of thransverse axis
$=2\sqrt{2}c=8\sqrt{2}$ unit