A is a point on x- axis, and point B (5,-4) and AB = 5 units find the co-ordinates of A.

Answer

To find the coordinates of point A on the x-axis such that the distance AB = 5 units, where B has coordinates (5, -4), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Coordinates of A**:
Since point A is on the x-axis, its coordinates can be represented as (x, 0), where x is the unknown we need to find.

2. **Use the Distance Formula**:
The distance between two points A(x1, y1) and B(x2, y2) is given by the formula:
\[ AB = \sqrt{(x2 - x1)^2 + (y2 - y1)^2} \]
Here, A is (x, 0) and B is (5, -4). So, we can substitute these values into the formula.

3. **Set Up the Equation**:
We know that the distance AB = 5 units, so we can set up the equation:
\[ 5 = \sqrt{(5 - x)^2 + (-4 - 0)^2} \]

4. **Square Both Sides**:
To eliminate the square root, we square both sides of the equation:
\[ 25 = (5 - x)^2 + (-4)^2 \]

5. **Simplify the Equation**:
Calculate (-4)^2:
\[ 25 = (5 - x)^2 + 16 \]
Now, subtract 16 from both sides:
\[ 25 - 16 = (5 - x)^2 \]
\[ 9 = (5 - x)^2 \]

6. **Take the Square Root**:
Taking the square root of both sides gives us:
\[ \sqrt{9} = |5 - x| \]
This results in two equations:
\[ 5 - x = 3 \quad \text{or} \quad 5 - x = -3 \]

7. **Solve for x**:
- For the first equation:
\[ 5 - x = 3 \implies x = 5 - 3 = 2 \]
- For the second equation:
\[ 5 - x = -3 \implies x = 5 + 3 = 8 \]

8. **Determine the Coordinates of A**:
Thus, the coordinates of point A can be:
- A(2, 0)
- A(8, 0)

### Final Answer:
The coordinates of point A are (2, 0) and (8, 0).