GATE CE 2023 Set 2 — Question 5
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General Aptitude → Spatial Aptitude → Transformation of Shapes (Translation, Rotation, Scaling, Mirroring)
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Question
Consider a circle with its centre at the origin , as shown. Two operations are allowed on the circle.Operation 1: Scale independently along the and axes.
Operation 2: Rotation in any direction about the origin.Which figure among the options can be achieved through a combination of these two operations on the given circle?
Operation 2: Rotation in any direction about the origin.Which figure among the options can be achieved through a combination of these two operations on the given circle?
Correct answer
(A) [figure]
Solution
1.Operation 1 (Scaling): Scaling a circle centered at the origin independently along the and axes transforms it into an ellipse centered at the origin. The equation changes from to . Crucially, the center of the figure remains at the origin .
2.Operation 2 (Rotation): Rotating any figure about the origin preserves the position of the origin. If a figure is centered at the origin, it will remain centered at the origin after any rotation about the origin.
3.Analysis of Options:
- Option (A) shows an ellipse whose center is at the origin . This can be achieved by first scaling the circle into an axis-aligned ellipse and then rotating it.
- Options (B), (C), and (D) show ellipses that are not centered at the origin (they have been translated/shifted). Since neither scaling along the axes nor rotation about the origin allows for translation, these figures cannot be achieved from the initial circle.
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