Question

The coordinate of the vertices of the rectangular hyperbola $xy=16$ are:

• A. $(4,0),(-4,0)$
• B. $(4,4),(-4,-4)$
• C. $(8,0),(-8,0)$
• D. $(2,8),(-2,-8)$

Answer 1

The correct option is B $(4,4),(-4,-4)$

General form of a rectangular hyperbola
$xy=c^2$ where $c^2=\frac{a^2}{2}.$
The equation of the given rectangular hyperbola is $xy=16$

$\Rightarrow$ $c^2=16$

$\Rightarrow$ $\frac{a^2}{2}=16$

$\Rightarrow$ $a^2=32$

$\Rightarrow$ $a=\sqrt{32}$

$\Rightarrow$ $a=\sqrt{16\times 2}$

$\therefore$ $a=4\sqrt{2}$

$\therefore$ Vertices are $A(\frac{4\sqrt{2}}{\sqrt{2}},\frac{4\sqrt{2}}{\sqrt{2}}),$ $A^{\prime }(\frac{-4\sqrt{2}}{\sqrt{2}},\frac{-4\sqrt{2}}{\sqrt{2}})$

$\therefore$ Vertices are $(4,4,),$ $(-4,-4)$