If z is a comlex number in the argand plane, the equation `|z-2|+|z+2|=8` represents

Options

A. a parabola
B. an ellipse
C. a hyperbola
D. a circle

Answer

To solve the equation \( |z - 2| + |z + 2| = 8 \) and determine what it represents in the Argand plane, we can follow these steps:

### Step 1: Understand the components of the equation
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. The expression \( |z - 2| \) represents the distance from the point \( z \) to the point \( 2 \) on the real axis (which is \( (2, 0) \) in the Argand plane), and \( |z + 2| \) represents the distance from \( z \) to the point \( -2 \) (which is \( (-2, 0) \)).

### Step 2: Rewrite the equation
The equation can be rewritten as:
\[ |z - 2| + |z + 2| = 8 \]
This means the sum of the distances from point \( z \) to the points \( 2 \) and \( -2 \) is equal to \( 8 \).

### Step 3: Identify the geometric representation
The equation \( |z - 2| + |z + 2| = 8 \) represents an ellipse in the Argand plane, where the foci are the points \( 2 \) and \( -2 \). The sum of the distances from any point on the ellipse to the two foci is constant (in this case, \( 8 \)).

### Step 4: Determine the characteristics of the ellipse
To confirm that it is indeed an ellipse, we need to check if the distance between the foci is less than the constant sum of distances. The distance between the foci \( 2 \) and \( -2 \) is:
\[ d = |2 - (-2)| = |2 + 2| = 4 \]
Since \( 4 < 8 \), this confirms that the shape is an ellipse.

### Step 5: Conclusion
Thus, the equation \( |z - 2| + |z + 2| = 8 \) represents an ellipse in the Argand plane with foci at \( 2 \) and \( -2 \).

### Summary of the solution
The equation \( |z - 2| + |z + 2| = 8 \) represents an ellipse in the Argand plane with foci at \( (2, 0) \) and \( (-2, 0) \).

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