Question

The point P(4, 1) undergoes the following transformations in succession: (i) origin is shifted to the point (1, 6) by translation of axes, (ii) translation through a distance of 2 units along the positive direction of the x-axis, (iii) rotation of axes through an angle of $90^\circ$ in the positive direction. Then the coordinates of the point P in its final position are

• A. (3, 4)
• B. (4, 3)
• C. (-5, -5)
• D. (1, 0)

Hint:

For axes rotation by $\theta$ counterclockwise, the point’s coordinates transform as $(x', y') = (y, -x)$ for $\theta = 90^\circ$. Apply transformations sequentially, tracking coordinates carefully.

Answer 1

The Correct Option is C

Apply the transformations in order: 1. Shift origin to (1, 6): New coordinates of $P(4, 1)$ are: \[ (x', y') = (4 - 1, 1 - 6) = (3, -5) \] 2. Translate 2 units along positive x-axis: In the new axes, move $P'(3, -5)$: \[ (x", y") = (3 + 2, -5) = (5, -5) \] 3. Rotate axes by $90^\circ$ counterclockwise: For a point $(x, y)$ in the original axes, if axes are rotated counterclockwise by $90^\circ$, the new coordinates $(x"', y"')$ relative to the rotated axes are found by transforming $(x, y)$ as if the point rotated clockwise by $90^\circ$ (since rotating axes counterclockwise is equivalent to rotating the point clockwise): \[ (x"', y"') = (y", -x") \] For $P"(5, -5)$: \[ (x"', y"') = (-5, -5) \] The final coordinates are $(-5, -5)$. Option (3) is correct. The original solution’s ambiguity about “rotation of axes” vs. “point rotation” is resolved by interpreting it as axes rotation. Options (1), (2), and (4) do not match the computed coordinates.