Question

The image of a point \( (2, -1) \) with respect to the line \( x - y + 1 = 0 \) is

• A. \( (2, -3) \)
• B. \( (-2, 3) \)
• C. \( (0, 1) \)
• D. \( (-1, 0) \)

Hint:

To find the reflection of a point about a line \( Ax + By + C = 0 \), use the standard formula involving the perpendicular projection distance and components.

Answer 1

The Correct Option is B

Step 1: Given point \( P(2, -1) \), and the line is \( x - y + 1 = 0 \) Step 2: Use the formula for reflection of a point \( (x_1, y_1) \) about the line \( Ax + By + C = 0 \): \[ (x', y') = \left( x_1 - \frac{2A(Ax_1 + By_1 + C)}{A^2 + B^2}, \ y_1 - \frac{2B(Ax_1 + By_1 + C)}{A^2 + B^2} \right) \] Step 3: For line \( x - y + 1 = 0 \), \( A = 1, B = -1, C = 1 \), and point \( (x_1, y_1) = (2, -1) \) \[ Ax_1 + By_1 + C = 1(2) + (-1)(-1) + 1 = 2 + 1 + 1 = 4 \] \[ A^2 + B^2 = 1^2 + (-1)^2 = 2 \] Step 4: Plug into the formula: \[ x' = 2 - \frac{2(1)(4)}{2} = 2 - 4 = -2 \] \[ y' = -1 - \frac{2(-1)(4)}{2} = -1 + 4 = 3 \] So the image point is \( (-2, 3) \).