Step-by-Step Solution
Key Concept: The locus $|z-z_1|+|z-z_2|=k$ (constant) defines an ellipse with foci at $z_1$ and $z_2$ when $k$ exceeds the distance between foci.
The equation $|z-i|+|z+i|=k$ represents the sum of distances from point $z$ to two fixed points $F_1 = i$ and $F_2 = -i$. By definition, when the sum of distances from a point to two fixed points (foci) is constant, the locus is an ellipse. Here the foci are at $(0,1)$ and $(0,-1)$ on the imaginary axis, separated by distance $2c=2$, so $c=1$. For an ellipse, we need $k > 2c = 2$. The major axis lies along the imaginary axis with semi-major axis $a = k/2$ and semi-minor axis $b = \sqrt{a^2-c^2} = \sqrt{(k/2)^2-1}$.
Correct Answer: [A-r] [B-s] [C-q] [D-p]