Find a relation between x and y such that the point (x, y) is equidistant from the `C(7,1)` and `D(2,3)`:

Options

A. x -y = 2
B. x + y = 2
C. 2x - y = 0
D. 2x + y = 2

Answer

To find a relation between \( x \) and \( y \) such that the point \( (x, y) \) is equidistant from the points \( C(7, 1) \) and \( D(2, 3) \), we can follow these steps:

### Step 1: Use the Distance Formula
The distance 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 Distances
We need to find the distance from point \( (x, y) \) to point \( C(7, 1) \) and point \( D(2, 3) \).

1. Distance from \( (x, y) \) to \( C(7, 1) \):
\[ d_{C} = \sqrt{(x - 7)^2 + (y - 1)^2} \]

2. Distance from \( (x, y) \) to \( D(2, 3) \):
\[ d_{D} = \sqrt{(x - 2)^2 + (y - 3)^2} \]

### Step 3: Set the Distances Equal
Since the point \( (x, y) \) is equidistant from \( C \) and \( D \), we can set the two distances equal to each other:
\[ \sqrt{(x - 7)^2 + (y - 1)^2} = \sqrt{(x - 2)^2 + (y - 3)^2} \]

### Step 4: Square Both Sides
To eliminate the square roots, we square both sides of the equation:
\[ (x - 7)^2 + (y - 1)^2 = (x - 2)^2 + (y - 3)^2 \]

### Step 5: Expand Both Sides
Now we expand both sides:
1. Left side:
\[ (x - 7)^2 + (y - 1)^2 = (x^2 - 14x + 49) + (y^2 - 2y + 1) = x^2 + y^2 - 14x - 2y + 50 \]

2. Right side:
\[ (x - 2)^2 + (y - 3)^2 = (x^2 - 4x + 4) + (y^2 - 6y + 9) = x^2 + y^2 - 4x - 6y + 13 \]

### Step 6: Set the Expanded Forms Equal
Now we set the expanded forms equal to each other:
\[ x^2 + y^2 - 14x - 2y + 50 = x^2 + y^2 - 4x - 6y + 13 \]

### Step 7: Simplify the Equation
We can cancel \( x^2 \) and \( y^2 \) from both sides:
\[ -14x - 2y + 50 = -4x - 6y + 13 \]

Now, rearranging gives:
\[ -14x + 4x + 6y - 2y + 50 - 13 = 0 \]
This simplifies to:
\[ -10x + 4y + 37 = 0 \]

### Step 8: Rearranging the Equation
Rearranging gives us the relation:
\[ 10x - 4y - 37 = 0 \]

### Final Relation
Thus, the relation between \( x \) and \( y \) such that the point \( (x, y) \) is equidistant from \( C(7, 1) \) and \( D(2, 3) \) is:
\[ 10x - 4y - 37 = 0 \]