Question

Point (-1, 2) is changed to (a, b) when the origin is shifted to the point (2, -1) by translation of axes. Point (a, b) is changed to (c, d) when the axes are rotated through an angle of 45$^{\circ}$ about the new origin. (c, d) is changed to (e, f) when (c, d) is reflected through y = x. Then (e, f) = ?

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

Hint:

Remember the formulas for translation and rotation of axes.

Answer 1

The Correct Option is C

We are required to follow three transformations: 1. Translation of axes 2. Rotation of axes by \( 45^\circ \) 3. Reflection through the line \(y = x\) Step 1: Translation of Axes 
The point \( (-1, 2) \) is translated when the origin is shifted to \( (2, -1) \). Using the translation formula: \[ x' = x - 2 \quad \text{and} \quad y' = y + 1 \] Substituting the given point: \[ a = -1 - 2 = -3 \quad \text{and} \quad b = 2 + 1 = 3 \] Thus, the new point is \( (-3, 3) \). 

Step 2: Rotation of Axes by \( 45^\circ \) 
The rotation transformation formula is: \[ x'' = x'\cos 45^\circ - y'\sin 45^\circ \] \[ y'' = x'\sin 45^\circ + y'\cos 45^\circ \] Since \( \cos 45^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2} \), we have: \[ x'' = (-3)\frac{\sqrt{2}}{2} - (3)\frac{\sqrt{2}}{2} = -\frac{3\sqrt{2}}{2} - \frac{3\sqrt{2}}{2} = -3\sqrt{2} \] \[ y'' = (-3)\frac{\sqrt{2}}{2} + (3)\frac{\sqrt{2}}{2} = -\frac{3\sqrt{2}}{2} + \frac{3\sqrt{2}}{2} = 0 \] So the new point is \( (-3\sqrt{2}, 0) \). 

Step 3: Reflection through \( y = x \) 
The reflection transformation formula for reflection across \(y = x\) is: \[ x''' = y'' \quad \text{and} \quad y''' = x'' \] Since \(y'' = 0\) and \(x'' = -3\sqrt{2} \), the reflection gives: \[ e = 0 \quad \text{and} \quad f = -3\sqrt{2} \] 
Step 4: Final Answer 
\[ \boxed{(3\sqrt{2}, 0)} \] 

Final Answer: (C) \( (3\sqrt{2}, 0) \)