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

Options

A. circle
B. pair of a 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 it step by step.

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

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

### Step 3: Apply the Distance Formula
The distance from point \( P(x, y) \) to point \( A(2, 0) \) is given by:
\[ PA = \sqrt{(x - 2)^2 + (y - 0)^2} = \sqrt{(x - 2)^2 + y^2} \]
Similarly, the distance from point \( P(x, y) \) to point \( B(-2, 0) \) is:
\[ PB = \sqrt{(x + 2)^2 + (y - 0)^2} = \sqrt{(x + 2)^2 + y^2} \]

### Step 4: Set Up the Equation
According to the problem, the sum of these distances is equal to 4:
\[ PA + PB = 4 \]

### Step 5: Interpret the Geometric Meaning
The equation \( PA + PB = d \) (where \( d = 4 \)) represents an ellipse with foci at points \( A(2, 0) \) and \( B(-2, 0) \). However, since the sum of distances is equal to the distance between the two foci (which is 4 in this case), it indicates that the points \( P(x, y) \) lie on the line segment connecting points \( A \) and \( B \).

### Step 6: Conclusion
Thus, the equation represents a line segment between the points \( A(2, 0) \) and \( B(-2, 0) \).

### Final Answer
The equation \( \sqrt{(x-2)^2 + y^2} + \sqrt{(x+2)^2 + y^2} = 4 \) represents a line segment between the points \( (2, 0) \) and \( (-2, 0) \).

---