The equation |z+i|-|z-i|=k represent a hyperbola, if

Step-by-step solution

Let z=x+i y , then |z+i|-|z-i|=k becomes x^2+(y+1)^2 - x^2+(y-1)^2 =k...(i) or x^2+(y+1)^2-x^2-(y-1)^2=k x^2+(y+1)^2 + x^2+(y-1)^2 x^2+(y-1)^2 + x^2+(y+1)^2 = 4 y k ...(ii) From Eqs. (i) and (ii), 2 x^2+(y+1)^2 =k+ 4 y k 4 x^2+4 y^2+8 y+4=k^2+ 16 y^2 k^2 +8 y 4 x^2+ (4- 16 k^2 ) y^2=k^2-4 For an hyperbola, 4 k^2-16 k^2 0 k^2-4 0 |k| 2 or -2 k 2