The coordinates of the point Q symmetric to the point P(–5, 13) with respect to the line 2x – 3y – 3 = 0 are -

Options

A. (11, –11)
B. (5, –13)
C. (7, –9)
D. (6, –3)

Answer

To find the coordinates of the point Q symmetric to the point P(–5, 13) with respect to the line 2x – 3y – 3 = 0, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given point and line**:
- Point P is given as \( P(-5, 13) \).
- The line is given as \( 2x - 3y - 3 = 0 \).

2. **Rewrite the line in the standard form**:
- The line can be expressed in the form \( Ax + By + C = 0 \) where \( A = 2 \), \( B = -3 \), and \( C = -3 \).

3. **Use the formula for the coordinates of the symmetric point**:
- The formula to find the coordinates of the point Q symmetric to P with respect to the line \( Ax + By + C = 0 \) is:
\[ Q(x, y) = \left( x_1 - \frac{2A(Ax_1 + By_1 + C)}{A^2 + B^2}, y_1 - \frac{2B(Ax_1 + By_1 + C)}{A^2 + B^2} \right) \]
- Here, \( (x_1, y_1) = (-5, 13) \).

4. **Calculate \( Ax_1 + By_1 + C \)**:
- Substitute \( x_1 = -5 \) and \( y_1 = 13 \):
\[ Ax_1 + By_1 + C = 2(-5) + (-3)(13) - 3 = -10 - 39 - 3 = -52 \]

5. **Calculate \( A^2 + B^2 \)**:
- Calculate:
\[ A^2 + B^2 = 2^2 + (-3)^2 = 4 + 9 = 13 \]

6. **Substitute values into the formula**:
- Now substitute \( A \), \( B \), \( C \), \( x_1 \), and \( y_1 \) into the formula for Q:
\[ Q(x, y) = \left( -5 - \frac{2 \cdot 2 \cdot (-52)}{13}, 13 - \frac{2 \cdot (-3) \cdot (-52)}{13} \right) \]

7. **Calculate the x-coordinate of Q**:
- Calculate:
\[ Q_x = -5 - \frac{-208}{13} = -5 + 16 = 11 \]

8. **Calculate the y-coordinate of Q**:
- Calculate:
\[ Q_y = 13 - \frac{312}{13} = 13 - 24 = -11 \]

9. **Final coordinates of Q**:
- Therefore, the coordinates of the point Q are \( Q(11, -11) \).

### Final Answer:
The coordinates of the point Q symmetric to the point P(–5, 13) with respect to the line \( 2x - 3y - 3 = 0 \) are \( (11, -11) \).