Question

The point \(P(\alpha, \beta)\) (\(\alpha>0, \beta>0\)) undergoes the following transformations successively.
a) Translation to a distance of 3 units in positive direction of x-axis. 
b) Reflection about the line \(y=-x\). 
c) Rotation of axes through an angle of \(\frac{\pi}{4}\) about the origin in the positive direction. 
If the final position of that point P is \((-4\sqrt{2}, -2\sqrt{2})\), then \((\alpha + \beta) =\)

• A. 5
• B. 7
• C. \(6\sqrt{2}\)
• D. \(2\sqrt{2}\)

Hint:

Distinguish between "Rotation of a point" and "Rotation of axes". 1. Rotation of point P to P' (axes fixed): New coords are rotated. 2. Rotation of axes (point P fixed): The coordinates of the same point change. Formula for axes rotation: \(x = X\cos\theta - Y\sin\theta\).

Answer 1

The Correct Option is A

Step 1: Apply Translation:
Original point \(P = (\alpha, \beta)\). Translation by 3 units in positive x-direction: \[ P_1 = (\alpha + 3, \beta) \]
Step 2: Apply Reflection:
Reflection of point \((x, y)\) about the line \(y = -x\) transforms it to \((-y, -x)\). \[ P_2 = (-\beta, -(\alpha + 3)) \] Let these coordinates in the original xy-system be denoted as \((x_{old}, y_{old}) = (-\beta, -\alpha - 3)\). 
Step 3: Apply Rotation of Axes:
The axes are rotated by \(\theta = 45^\circ\) (\(\pi/4\)). The coordinates of the fixed point \(P_2\) in the new system are given as \((X, Y) = (-4\sqrt{2}, -2\sqrt{2})\). The relation between old coordinates \((x, y)\) and new coordinates \((X, Y)\) when axes are rotated by \(\theta\) is: \[ x = X\cos\theta - Y\sin\theta \] \[ y = X\sin\theta + Y\cos\theta \] Substitute \(X = -4\sqrt{2}, Y = -2\sqrt{2}\) and \(\theta = 45^\circ\) (\(\cos45^\circ = \sin45^\circ = \frac{1}{\sqrt{2}}\)): \[ x_{old} = (-4\sqrt{2})\left(\frac{1}{\sqrt{2}}\right) - (-2\sqrt{2})\left(\frac{1}{\sqrt{2}}\right) = -4 - (-2) = -2 \] \[ y_{old} = (-4\sqrt{2})\left(\frac{1}{\sqrt{2}}\right) + (-2\sqrt{2})\left(\frac{1}{\sqrt{2}}\right) = -4 - 2 = -6 \] 
Step 4: Solve for \(\alpha\) and \(\beta\):
Equating the coordinates from Step 2 to the values from Step 3: \[ -\beta = -2 \implies \beta = 2 \] \[ -(\alpha + 3) = -6 \implies \alpha + 3 = 6 \implies \alpha = 3 \] Both \(\alpha\) and \(\beta\) are positive, satisfying the condition. 
Step 5: Calculate Sum:
\[ \alpha + \beta = 3 + 2 = 5 \]