The equation $\sqrt{x^2 + (y-1)^2} - \sqrt{x^2 + (y+1)^2} = K$ will represent a hyperbola for
Step-by-Step Solution
Key Concept: The equation $\sqrt{x^2 + (y-1)^2} - \sqrt{x^2 + (y+1)^2} = K$ represents the locus of points where the difference of distances from foci $F_1=(0,1)$ and $F_2=(0,-1)$ is constant; for a hyperbola, we require $0 < |K| < 2a$ where $2a$ is the distance between foci, giving $0 < |K| < 2$.
The equation $\sqrt{x^2+(y-1)^2}-\sqrt{x^2+(y+1)^2}=K$ represents a hyperbola where the difference of distances from points $S_1=(0,1)$ and $S_2=(0,-1)$ is constant. With $2a=K$ and $2ae=S_1S_2=2$, we get $e=\frac{2}{K}$. Since $e>1$ for a hyperbola, we need $K1$ is required for hyperbolas, missing the upper bound on $K$.
Correct Answer: 1