Answer :

Answer:

To show that the locus of a variable point \( p \) (represented by a complex number \( z \)) subject to the condition \( |z + 1| + |z - 1| = 3 \) describes an ellipse, we can follow these steps:

1. **Understand the Condition**:

  The given condition \( |z + 1| + |z - 1| = 3 \) represents the sum of the distances from the point \( z \) to the points \( -1 \) and \( 1 \).

2. **Foci of the Ellipse**:

  Recall that the definition of an ellipse is the set of points where the sum of the distances to two fixed points (foci) is a constant. Here, the fixed points (foci) are \( -1 \) and \( 1 \) on the real axis, and the constant sum is 3.

3. **Check the Major Axis Length**:

  For an ellipse with foci at \( -1 \) and \( 1 \), the sum of the distances to the foci equals the major axis length \( 2a \). So, \( 2a = 3 \), giving \( a = 1.5 \).

4. **Determine the Distance Between the Foci**:

  The distance between the foci is \( 2c \). Since the foci are \( -1 \) and \( 1 \), the distance between them is \( 2 \), so \( c = 1 \).

5. **Find the Minor Axis Length**:

  The relationship between \( a \), \( b \) (the semi-minor axis), and \( c \) in an ellipse is given by \( c^2 = a^2 - b^2 \). Substituting the known values:

  \[

  1^2 = 1.5^2 - b^2

  \]

  \[

  1 = 2.25 - b^2

  \]

  \[

  b^2 = 2.25 - 1 = 1.25

  \]

  \[

  b = \sqrt{1.25} = \frac{\sqrt{5}}{2}

  \]

Thus, the ellipse has a semi-major axis \( a = 1.5 \) and a semi-minor axis \( b = \frac{\sqrt{5}}{2} \).

**To sketch the ellipse:**

1. Draw the real axis and mark the foci at \( -1 \) and \( 1 \).

2. Draw the major axis with a total length of 3 units, centered at the origin.

3. Draw the minor axis perpendicular to the major axis at the center, with a total length of \( \sqrt{5} \) units.

Here's a sketch of the ellipse:

```plaintext

y-axis

^

|

|           .             .  (points on the ellipse)

|        .     .

|     .           .

|    .             .

|   .               .

| .                 .

|.                   .

----------------------------- -> x-axis

-2  -1    0    1     2

```

To accurately draw this:

- The center is at the origin (0,0).

- The foci are at \((-1, 0)\) and \( (1, 0) \).

- The vertices on the major axis are at \((\pm 1.5, 0)\).

- The vertices on the minor axis are at \((0, \pm \frac{\sqrt{5}}{2})\).

This sketch represents an ellipse centered at the origin with the major axis along the x-axis and the minor axis along the y-axis.