Question

The center of a rectangular hyperbola lies on the line $y=2x$. If one of the asymptotes is $x+y+c=0$, then the other asymptote is

• A. $x-y-3c=0$
• B. $2x-y+c=0$
• C. $x-y-c=0$
• D. none of these

Answer 1

The correct option is D none of these

The asymptotes of a rectangular hyperbola are perpendicular to each other.
Given one asymptote
$x+y+c=0$
Let the other asymptote be
$x-y+\lambda =0$
We also know that the asymptotes pass through the center of the hyperbola. Therefore, the line $2x-y=0$ and the asymptotes must be concurrent.
Thus, we have
$\left|\begin{array}{ccc}2& -1& 0\\ 1& 1& c\\ 1& -1& \lambda \end{array}\right|=0$
or $\lambda =-\frac{c}{3}$