Question

$ABCD$ is a square with $A=(-4,0),B=(4,0)$ and other vertices of the square lie above the $x-$axis. Let $O$ be the origin and $O_1$ be the mid point of $CD$. For a rectangular hyperbola if one branch passes through the points $C,D$, other branch passes through origin and its transverse axis is along the straight line $O{O}_1$ ,then :

• A. Centre of hyperbola is $(0,3)$
• B. One of the asymptotes of the hyperbola is $y=x+3$
• C. Centre of hyperbola is $(0,4)$
• D. One of the asymptotes of the hyperbola is $y=x+4$

Answer 1

Answer: (A), (B)

The correct options are
A Centre of hyperbola is $(0,3)$
B One of the asymptotes of the hyperbola is $y=x+3$


Let center of hyperbola be $(0,k)$
Let the equation of rectangular hyperbola be whose transverse axis along $y$-axis is :
$\Rightarrow (y-k{)}^2-(x{)}^2=a^2$
It passes through $(0,0)$ and $(-4,8)$
Hence we get $k=a$
$\Rightarrow (y-k{)}^2-x^2=k^2$
Putting $(-4,8)$ we get
$\Rightarrow (8-k{)}^2-16=k^2$
$\therefore k=3$
Asymptotes of hyperbola is $(y-k)=\pm (x-h)$
$\therefore y=3-x$ and $y=x+3$