If the point (x, y) is equidistant from points (7, 1) and (3, 5) , find x-y.

Options

A. 2
B. 4
C. 6
D. 8

Answer

To solve the problem, we need to find the point (x, y) that is equidistant from the points (7, 1) and (3, 5). We will use the distance formula to set up an equation and then solve for x and y.

### Step-by-Step Solution:

1. **Set Up the Distance Equation**:
Since the point (x, y) is equidistant from (7, 1) and (3, 5), we can express this using the distance formula:
\[ \sqrt{(x - 7)^2 + (y - 1)^2} = \sqrt{(x - 3)^2 + (y - 5)^2} \]

2. **Square Both Sides**:
To eliminate the square roots, we square both sides of the equation:
\[ (x - 7)^2 + (y - 1)^2 = (x - 3)^2 + (y - 5)^2 \]

3. **Expand Both Sides**:
Expanding both sides gives us:
\[ (x^2 - 14x + 49) + (y^2 - 2y + 1) = (x^2 - 6x + 9) + (y^2 - 10y + 25) \]

4. **Combine Like Terms**:
Simplifying both sides results in:
\[ x^2 + y^2 - 14x - 2y + 50 = x^2 + y^2 - 6x - 10y + 34 \]

5. **Cancel Common Terms**:
We can cancel \(x^2\) and \(y^2\) from both sides:
\[ -14x - 2y + 50 = -6x - 10y + 34 \]

6. **Rearrange the Equation**:
Rearranging gives us:
\[ -14x + 6x + 10y - 2y + 50 - 34 = 0 \]
Simplifying further:
\[ -8x + 8y + 16 = 0 \]

7. **Divide by -8**:
Dividing the entire equation by -8 gives:
\[ x - y - 2 = 0 \]

8. **Solve for y**:
Rearranging gives us:
\[ y = x - 2 \]

9. **Find x - y**:
We need to find \(x - y\):
\[ x - y = x - (x - 2) = 2 \]

### Final Answer:
Thus, \(x - y = 2\).

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