Find the retation between x and y, If the point (x,y) is to be equidistant from (6,-1) and (2,3)

Answer

To find the relation between \( x \) and \( y \) such that the point \( (x, y) \) is equidistant from the points \( (6, -1) \) and \( (2, 3) \), we can follow these steps:

### Step 1: Use the Distance Formula
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Step 2: Set Up the Equations
We need to find the distance from \( (x, y) \) to \( (6, -1) \) and from \( (x, y) \) to \( (2, 3) \). Thus, we have:
\[ \text{Distance from } (x, y) \text{ to } (6, -1) = \sqrt{(x - 6)^2 + (y + 1)^2} \]
\[ \text{Distance from } (x, y) \text{ to } (2, 3) = \sqrt{(x - 2)^2 + (y - 3)^2} \]

### Step 3: Set the Distances Equal
Since the point \( (x, y) \) is equidistant from both points, we can set the two distances equal to each other:
\[ \sqrt{(x - 6)^2 + (y + 1)^2} = \sqrt{(x - 2)^2 + (y - 3)^2} \]

### Step 4: Square Both Sides
To eliminate the square roots, we square both sides:
\[ (x - 6)^2 + (y + 1)^2 = (x - 2)^2 + (y - 3)^2 \]

### Step 5: Expand Both Sides
Now we expand both sides:
\[ (x^2 - 12x + 36) + (y^2 + 2y + 1) = (x^2 - 4x + 4) + (y^2 - 6y + 9) \]

### Step 6: Simplify the Equation
Combine like terms:
\[ x^2 - 12x + 36 + y^2 + 2y + 1 = x^2 - 4x + 4 + y^2 - 6y + 9 \]
The \( x^2 \) and \( y^2 \) terms cancel out:
\[ -12x + 36 + 2y + 1 = -4x + 4 - 6y + 9 \]

### Step 7: Rearrange the Equation
Rearranging gives:
\[ -12x + 36 + 2y + 1 + 4x + 6y - 9 = 0 \]
Combine like terms:
\[ -8x + 8y + 28 = 0 \]

### Step 8: Solve for the Relation
Rearranging the equation gives:
\[ -8x + 8y = -28 \]
Dividing through by -8:
\[ x - y = 3 \]

### Final Relation
Thus, the relation between \( x \) and \( y \) is:
\[ x - y = 3 \]