Find the relation between x and y if A(x, y), B(-2, 3) and C(2, 1) from an isosceles triangle with AB=AC.

Answer

To find the relation between \( x \) and \( y \) such that the points \( A(x, y) \), \( B(-2, 3) \), and \( C(2, 1) \) form an isosceles triangle with \( AB = AC \), 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:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
We will use this formula to find the lengths \( AB \) and \( AC \).

### Step 2: Calculate \( AB \)
Using the distance formula, the distance \( AB \) is:
\[ AB = \sqrt{((-2) - x)^2 + (3 - y)^2} \]

### Step 3: Calculate \( AC \)
Similarly, the distance \( AC \) is:
\[ AC = \sqrt{(2 - x)^2 + (1 - y)^2} \]

### Step 4: Set the Distances Equal
Since we know that \( AB = AC \), we can set the two expressions equal to each other:
\[ \sqrt{((-2) - x)^2 + (3 - y)^2} = \sqrt{(2 - x)^2 + (1 - y)^2} \]

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

### Step 6: Expand Both Sides
Expanding the left side:
\[ (-2 - x)^2 = (x + 2)^2 = x^2 + 4 + 4x \]
\[ (3 - y)^2 = 9 - 6y + y^2 \]
So, the left side becomes:
\[ x^2 + 4 + 4x + 9 - 6y + y^2 = x^2 + y^2 + 4x - 6y + 13 \]

Expanding the right side:
\[ (2 - x)^2 = (x - 2)^2 = x^2 - 4 + 4x \]
\[ (1 - y)^2 = 1 - 2y + y^2 \]
So, the right side becomes:
\[ x^2 - 4 + 4x + 1 - 2y + y^2 = x^2 + y^2 + 4x - 2y - 3 \]

### Step 7: Set the Expanded Forms Equal
Now we have:
\[ x^2 + y^2 + 4x - 6y + 13 = x^2 + y^2 + 4x - 2y - 3 \]

### Step 8: Simplify
Cancel \( x^2 \) and \( y^2 \) from both sides:
\[ 4x - 6y + 13 = 4x - 2y - 3 \]
Subtract \( 4x \) from both sides:
\[ -6y + 13 = -2y - 3 \]

### Step 9: Rearrange the Equation
Add \( 6y \) to both sides:
\[ 13 = 4y - 3 \]
Add \( 3 \) to both sides:
\[ 16 = 4y \]
Divide by \( 4 \):
\[ y = 4 \]

### Step 10: Substitute \( y \) Back
Now substituting \( y = 4 \) back into the relation:
From our earlier equation, we can express \( x \):
\[ 2x + 2 = 5 \]
Subtract \( 2 \):
\[ 2x = 3 \]
Divide by \( 2 \):
\[ x = \frac{3}{2} \]

### Final Relation
Thus, the relation between \( x \) and \( y \) is:
\[ x = \frac{3}{2}, \quad y = 4 \]