Question

If point $P(x,y)$ is such that it moves on a hyperbola $|\sqrt{(x-3{)}^2+(y-4{)}^2}-\sqrt{x^2+y^2}|=k^2+1$, then the number of possible integral value(s) of $k$ is

Answer 1

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$|\sqrt{(x-3{)}^2+(y-4{)}^2}-\sqrt{x^2+y^2}|=k^2+1$ represents a hyperbola,
if $k^2+1<AB$ where $A=(3,4),\text{and}B=(0,0)$
$\Rightarrow k^2<4$
Thus, $k$ can take $3$ integral values namely $-1,0,1$.