If A (x, y) is equidistant from P (-3, 2) and Q (2,-3), then

Options

A. `2x = y`
B. `x = -y`
C. `x = 2y`
D. `x = y`

Answer

To solve the problem where point A (x, y) is equidistant from points P (-3, 2) and Q (2, -3), we will use the distance formula and set the distances equal to each other.

### Step-by-Step Solution:

1. **Understand the concept of equidistance**:
- If point A is equidistant from points P and Q, then the distance from A to P (AP) is equal to the distance from A to Q (AQ).
- Mathematically, this can be expressed as:
\[ AP = AQ \]

2. **Use the distance formula**:
- The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
- For our points:
- Point P: \((-3, 2)\)
- Point Q: \((2, -3)\)
- Point A: \((x, y)\)

3. **Calculate the distance AP**:
- Using the distance formula:
\[ AP = \sqrt{(x - (-3))^2 + (y - 2)^2} \]
\[ AP = \sqrt{(x + 3)^2 + (y - 2)^2} \]

4. **Calculate the distance AQ**:
- Similarly, for AQ:
\[ AQ = \sqrt{(x - 2)^2 + (y - (-3))^2} \]
\[ AQ = \sqrt{(x - 2)^2 + (y + 3)^2} \]

5. **Set the distances equal**:
- Since A is equidistant from P and Q:
\[ \sqrt{(x + 3)^2 + (y - 2)^2} = \sqrt{(x - 2)^2 + (y + 3)^2} \]

6. **Square both sides to eliminate the square roots**:
- This gives:
\[ (x + 3)^2 + (y - 2)^2 = (x - 2)^2 + (y + 3)^2 \]

7. **Expand both sides**:
- Left-hand side:
\[ (x + 3)^2 = x^2 + 6x + 9 \]
\[ (y - 2)^2 = y^2 - 4y + 4 \]
\[ LHS = x^2 + 6x + 9 + y^2 - 4y + 4 = x^2 + y^2 + 6x - 4y + 13 \]

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

8. **Set the expanded equations equal**:
- Now we have:
\[ x^2 + y^2 + 6x - 4y + 13 = x^2 + y^2 - 4x + 6y + 13 \]

9. **Cancel out common terms**:
- We can cancel \(x^2\), \(y^2\), and \(13\) from both sides:
\[ 6x - 4y = -4x + 6y \]

10. **Rearrange the equation**:
- Combine like terms:
\[ 6x + 4x = 6y + 4y \]
\[ 10x = 10y \]

11. **Divide by 10**:
- This simplifies to:
\[ x = y \]

### Final Answer:
The point A (x, y) is such that \( x = y \).