Question

The equation $|\sqrt{(x-2{)}^2+(y-1{)}^2}-\sqrt{(x+2{)}^2+y^2}|=c$ will represent a hyperbola if

• A. $c\in (0,6)$
• B. $c\in (0,5)$
• C. $c\in (0,\sqrt{17})$
• D. $c\in (0,\sqrt{19})$

Answer 1

The correct option is C $c\in (0,\sqrt{17})$

In a plane if $A$ and $B$ are two fixed point and a variable point $P$ is moving such that $|PA-PB|=k,k<AB$, then it will represent a hyperbola having foci at $A,B$

$\therefore$ equation $|\sqrt{(x-2{)}^2+(y-1{)}^2}-\sqrt{(x+2{)}^2+y^2}|=c$ represnts hyperbola if $c<AB$
where $A=(2,1),\text{and}B=(-2,0)$
$\therefore c<\sqrt{17}$