Q. For a complex number `z` ,```|z-1|+|z+1| = 2` Then `z` lies on a A) parabola B) line segment C) circle D) none of these
You need to use the following identity, such that:
`|z - z_1| + |z - z_2| = c`
`z_1,z_2` represent two arbitrary fixed complex numbers
`c` represents a real number
You also should notice that the shape depends on the following triangle inequality `|z_1| + |z_2| > |z_1 - z_2|` and the following relations, such that: `c<|z_1 - z_2|, c=|z_1 - z_2|,c>|z_1 - z_2|` .
Identifying `z_1 = 1,z_2 = -1` and `c = 2` yields:
`|z_1 - z_2| = c => |1 - (-1)| = |1 + 1| = |2| = 2 = c`
Since the problem provides the relation `|z - 1| + |z + 1| = 2` and `|z_1 - z_2| = |2| = 2 = c` yields that z lies on a line segment that connects `z_1 = 1` and `z_2 = -1` .
Hence, evaluating the answer you need to select, yields that B) line segment is the valid answer.