Question

If the asymptotes of the hyperbola $(x+y+1{)}^2-(x-y-3{)}^2=5$
cut each other at $A$ and the coordinate axis at $B$ and $C$ then radius of circle passing through the points $A,B,C$ is :

• A. $\frac{\sqrt{5}}{2}$ unit
• B. $\frac{3}{2}$ unit
• C. $\frac{1}{2}$ unit
• D. $\frac{\sqrt{3}}{2}$ unit

Answer 1

The correct option is A $\frac{\sqrt{5}}{2}$ unit

Given equation of hyperbola is $(x+y+1{)}^2-(x-y-3{)}^2=5$
$\Rightarrow (2x-2)(2y+4)=5$
$\Rightarrow (x-1)(y+2)=\frac{5}{4}$
$\therefore$Centre is $(1,-2)$
$\because$ Equation of asymptotes for the hyperbola $(x-h)(y-k)=a^2$ are
$x=h$ and $y=k$
$\therefore x=1$ and $y=-2$ are the required asymptotes
Hence, we get $A(1,-2),B(1,0)$ and $C(0,-2)$
So, $△ABC$ will be rightangle triangle with $A$ as rightangle
Hence radius of circum-circle of $△ABC$ will be
$r=\frac{BC}{2}=\frac{\sqrt{5}}{2}$ unit