Write the coordinates of a point P on x-axis which is equidistant from the points A(-2, 0) and B(6, 0).

Answer

To find the coordinates of point P on the x-axis that is equidistant from points A(-2, 0) and B(6, 0), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
We need to find a point P on the x-axis such that the distance from P to A is equal to the distance from P to B. Since P is on the x-axis, its coordinates can be represented as P(x, 0).

2. **Set Up the Distance Formula**:
The distance between two points (x1, y1) and (x2, y2) is given by the formula:
\[ d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2} \]
For our case, we will set the distances PA and PB equal to each other:
\[ PA = PB \]

3. **Calculate Distance PA**:
The distance from P(x, 0) to A(-2, 0):
\[ PA = \sqrt{(x - (-2))^2 + (0 - 0)^2} = \sqrt{(x + 2)^2} \]
Since we are dealing with distances, we can drop the square root:
\[ PA = |x + 2| \]

4. **Calculate Distance PB**:
The distance from P(x, 0) to B(6, 0):
\[ PB = \sqrt{(x - 6)^2 + (0 - 0)^2} = \sqrt{(x - 6)^2} \]
Again, we can drop the square root:
\[ PB = |x - 6| \]

5. **Set the Distances Equal**:
Now, we set the two distances equal to each other:
\[ |x + 2| = |x - 6| \]

6. **Consider Cases for Absolute Values**:
We need to consider two cases based on the definition of absolute values.

**Case 1**: \(x + 2 = x - 6\)
\[ 2 = -6 \quad \text{(This case is not possible)} \]

**Case 2**: \(x + 2 = -(x - 6)\)
\[ x + 2 = -x + 6 \]
\[ 2x = 4 \quad \Rightarrow \quad x = 2 \]

7. **Find the Coordinates of P**:
Since we found \(x = 2\), the coordinates of point P are:
\[ P(2, 0) \]

### Final Answer:
The coordinates of point P on the x-axis that is equidistant from points A and B are \(P(2, 0)\).