Question

If \( Q \) and \( R \) are the images of the point \( P(2,3) \) with respect to the lines \( x - y + 2 = 0 \) and \( 2x + y - 2 = 0 \) respectively, then \( Q \) and \( R \) lie on

• A. the same side of the line \( 2x + y - 2 = 0 \)
• B. the opposite sides of the line \( 2x - y - 2 = 0 \)
• C. the same side of the line \( x + y + 2 = 0 \)
• D. the opposite sides of the line \( x - y + 2 = 0 \) \bigskip

Hint:

Use the reflection formula to systematically compute the image of a point with respect to a line.

Answer 1

The Correct Option is C

Let \( P(2, 3) \) be the given point. This point is reflected across the lines \( x - y + 2 = 0 \) and \( 2x + y - 2 = 0 \). We use the standard reflection formula to find the corresponding image points \( Q \) and \( R \). 1. The reflection formula for a point \( P(x_1, y_1) \) about a line \( Ax + By + C = 0 \) is: \[ x' = x_1 - \frac{2A(Ax_1 + By_1 + C)}{A^2+B^2}, \quad y' = y_1 - \frac{2B(Ax_1 + By_1 + C)}{A^2+B^2}. \] 2. Applying this formula to the line \( x - y + 2 = 0 \) gives the coordinates of the reflected point \( Q \). 3. Similarly, using the formula for the line \( 2x + y - 2 = 0 \) yields the coordinates of the reflected point \( R \). 4. Analyzing the positions of \( Q \) and \( R \) shows that both points lie on the same side of the line \( x + y + 2 = 0 \). \bigskip