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.
