Question

Number of complex numbers satisfying the relation $|z + \overline{z}| + |z - \overline{z}| = 2$ and $|z + i| + |z - i| = 2$, is

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (B)

2

Answer 2

Let $z = x + iy$, where $x$ and $y$ are real numbers. Then $\overline{z} = x - iy$.

We are given two conditions:

1. $|z + \overline{z}| + |z - \overline{z}| = 2$
2. $|z + i| + |z - i| = 2$

Let's analyze the first condition: $z + \overline{z} = (x + iy) + (x - iy) = 2x$ $z - \overline{z} = (x + iy) - (x - iy) = 2iy$

Substitute these into the first condition: $|2x| + |2iy| = 2$ $2|x| + 2|i||y| = 2$ Since $|i| = 1$: $2|x| + 2|y| = 2$ Dividing by 2, we get: $|x| + |y| = 1$

This equation represents a square in the complex plane (or xy-plane) with vertices at $(1,0)$, $(0,1)$, $(-1,0)$, and $(0,-1)$.

Now let's analyze the second condition: $|z + i| + |z - i| = 2$ This equation is of the form $|z - F_1| + |z - F_2| = 2a$, which describes the locus of a point $z$ such that the sum of its distances from two fixed points (foci $F_1$ and $F_2$) is a constant $2a$. In this case, the foci are $F_1 = -i$ and $F_2 = i$. The distance between the foci is $2c = |i - (-i)| = |2i| = 2$. The given sum of distances is $2a = 2$.

Since $2a = 2c$, the ellipse degenerates into the line segment connecting the two foci. Therefore, $z$ must lie on the line segment connecting $-i$ and $i$. This means $z$ must be a purely imaginary number $z = iy$, where the imaginary part $y$ is between $-1$ and $1$ (inclusive). So, $x = 0$ and $-1 \le y \le 1$.

Now we need to find the complex numbers $z = x + iy$ that satisfy both conditions. From the second condition, we know $x = 0$. Substitute $x = 0$ into the first condition: $|0| + |y| = 1$ $|y| = 1$ This implies $y = 1$ or $y = -1$.

Both $y = 1$ and $y = -1$ satisfy the range $-1 \le y \le 1$ obtained from the second condition. Thus, the possible values for $z$ are:

1. $x = 0, y = 1 \implies z = 0 + i(1) = i$
2. $x = 0, y = -1 \implies z = 0 + i(-1) = -i$

Let's verify these solutions:

For $z=i$:

1. $|i + \overline{i}| + |i - \overline{i}| = |i - i| + |i - (-i)| = |0| + |2i| = 0 + 2 = 2$. (Satisfied)
2. $|i + i| + |i - i| = |2i| + |0| = 2 + 0 = 2$. (Satisfied)

For $z=-i$:

1. $|-i + \overline{-i}| + |-i - \overline{-i}| = |-i + i| + |-i - (-i)| = |0| + |-2i| = 0 + 2 = 2$. (Satisfied)
2. $|-i + i| + |-i - i| = |0| + |-2i| = 0 + 2 = 2$. (Satisfied)

Both $z=i$ and $z=-i$ satisfy both given conditions. Therefore, there are 2 complex numbers satisfying the given relations.