The equation `sqrt((x-2)^(2)+y^(2))+sqrt((x+2)^(2)+y^(2))=4` represents

Options

A. a circle
B. a pair of lines
C. a parabola
D. an ellipse

Answer

To solve the equation \( \sqrt{(x-2)^{2} + y^{2}} + \sqrt{(x+2)^{2} + y^{2}} = 4 \), we will analyze the geometric interpretation of the equation step by step.

### Step 1: Understand the Equation
The equation represents the sum of the distances from a point \( P(x, y) \) to two fixed points \( A(-2, 0) \) and \( B(2, 0) \). The left side of the equation is the sum of distances \( AP + BP \).

### Step 2: Identify the Fixed Points
The fixed points are:
- Point A: \( (-2, 0) \)
- Point B: \( (2, 0) \)

### Step 3: Analyze the Sum of Distances
The equation states that the sum of the distances from point \( P \) to points \( A \) and \( B \) is equal to 4. This is a specific case of the definition of an ellipse, where the sum of the distances from any point on the ellipse to the two foci (in this case, points \( A \) and \( B \)) is constant.

### Step 4: Determine the Length of the Major Axis
The distance between points \( A \) and \( B \) is:
\[ AB = |2 - (-2)| = 4 \]
Since the sum of the distances \( AP + BP = 4 \) is equal to the distance \( AB \), this indicates that the locus of point \( P \) does not form an ellipse but rather a degenerate case where the locus is a line segment between points \( A \) and \( B \).

### Step 5: Conclusion
Since the sum of the distances equals the distance between the two points, the locus of point \( P \) is the line segment connecting points \( A \) and \( B \). Therefore, the equation represents a pair of lines.

### Final Answer
The equation \( \sqrt{(x-2)^{2} + y^{2}} + \sqrt{(x+2)^{2} + y^{2}} = 4 \) represents a pair of lines.

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