Find the equation to the locus of points equidistant from the points
(a + b,a - b),(a - b,a + b)

Answer

To find the equation of the locus of points equidistant from the points \( (a + b, a - b) \) and \( (a - b, a + b) \), we can follow these steps:

### Step 1: Define the points and the variable point
Let the two points be:
- Point A: \( (a + b, a - b) \)
- Point B: \( (a - b, a + b) \)

Let the point \( P \) be \( (x, y) \).

### Step 2: Use the distance formula
We need to find the distances from point \( P \) to points \( A \) and \( B \) and set them equal since the locus consists of points equidistant from \( A \) and \( B \).

The distance \( PA \) from point \( P \) to point \( A \) is given by:
\[ PA = \sqrt{(x - (a + b))^2 + (y - (a - b))^2} \]

The distance \( PB \) from point \( P \) to point \( B \) is given by:
\[ PB = \sqrt{(x - (a - b))^2 + (y - (a + b))^2} \]

### Step 3: Set the distances equal
Since \( PA = PB \), we have:
\[ \sqrt{(x - (a + b))^2 + (y - (a - b))^2} = \sqrt{(x - (a - b))^2 + (y - (a + b))^2} \]

### Step 4: Square both sides to eliminate the square roots
Squaring both sides gives:
\[ (x - (a + b))^2 + (y - (a - b))^2 = (x - (a - b))^2 + (y - (a + b))^2 \]

### Step 5: Expand both sides
Expanding both sides:
Left side:
\[ (x - (a + b))^2 + (y - (a - b))^2 = (x^2 - 2x(a + b) + (a + b)^2) + (y^2 - 2y(a - b) + (a - b)^2) \]
Right side:
\[ (x - (a - b))^2 + (y - (a + b))^2 = (x^2 - 2x(a - b) + (a - b)^2) + (y^2 - 2y(a + b) + (a + b)^2) \]

### Step 6: Simplify the equation
After expanding and simplifying both sides, we can cancel out \( x^2 \) and \( y^2 \) from both sides:
\[ -2x(a + b) - 2y(a - b) + (a + b)^2 + (a - b)^2 = -2x(a - b) - 2y(a + b) + (a - b)^2 + (a + b)^2 \]

### Step 7: Rearranging terms
Rearranging gives:
\[ -2x(a + b) + 2x(a - b) + 2y(a - b) - 2y(a + b) = 0 \]

### Step 8: Factor out common terms
Factoring out common terms leads to:
\[ 2x(b) - 2y(b) = 0 \]

### Step 9: Final equation
This simplifies to:
\[ x - y = 0 \]
or
\[ x = y \]

### Conclusion
The equation of the locus of points equidistant from the points \( (a + b, a - b) \) and \( (a - b, a + b) \) is:
\[ x - y = 0 \]