The equation | x^ 2 + (y - 1) ^ 2 - x^ 2 + (y + 1) ^ 2 | = K will...

We have \left|\sqrt{x^{2} + {\left(y - 1\right)}^{2}} - \sqrt{x^{2} + {\left(y + 1\right)}^{2}}\right| = K$<br /><br />Which is equivalent to \left|{S}_{1}P - {S}_{2}P\right|= Const .$

Where {S}_{1} ≡ \left(0, 1\right), {S}_{2} ≡ \left(0, -1\right)$.and$P ≡ \left(x, y\right)$<br /><br />Using properties of a hyperbola, the above equation represents a hyperbola, then we have.<br /><br />$2a = K$[where$2a$is the transverse axis and e is the eccentricity]<br /><br />and$2ae = {S}_{1}{S}_{2} = 2$<br /><br />Dividing, we have$e = \dfrac{2}{K}$<br /><br />Since,$e > 1$for a hyperbola, therefore$K < 2$<br /><br />Also,$K$must be a positive quantity. Hence, we have,$K \in $\left(0, 2\right)$ .