The equation `sqrt((x-4)^2+(y-2)^2)+sqrt((x+4)^2+(y-2)^2)=8` represents a

Explanation

To determine what the equation

\[ \sqrt{(x-4)^2+(y-2)^2} + \sqrt{(x+4)^2+(y-2)^2} = 8 \]

represents, we can follow these steps:

### Step 1: Identify the components of the equation
The equation consists of two square root terms, each representing the distance from a point \( P(x, y) \) to two fixed points \( A(4, 2) \) and \( B(-4, 2) \).

### Step 2: Recognize the geometric meaning
The equation states that the sum of the distances from point \( P \) to points \( A \) and \( B \) is constant (equal to 8). This is a defining property of an **ellipse** when the distance is less than the distance between the two foci. However, since the distance is equal to the distance between the foci, it indicates a special case.

### Step 3: Calculate the distance between the foci
To find the distance between points \( A \) and \( B \):

\[ AB = \sqrt{(4 - (-4))^2 + (2 - 2)^2} = \sqrt{(4 + 4)^2} = \sqrt{8^2} = 8 \]

### Step 4: Determine the type of conic section
Since the sum of the distances from any point \( P \) to the two foci \( A \) and \( B \) equals the distance \( AB \), this indicates that the curve is a **hyperbola**.

### Step 5: Conclusion
Thus, the equation

\[ \sqrt{(x-4)^2+(y-2)^2} + \sqrt{(x+4)^2+(y-2)^2} = 8 \]

represents a hyperbola with foci at \( A(4, 2) \) and \( B(-4, 2) \).