GATE 2023 MN – Question 5
Consider a circle with its centre at the origin (O), as shown. Two operations are allowed on the circle.
Operation 1: Scale independently along the x and y 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?
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Show answer and explanation
Correct answer: (A) Option A in the figure
Explanation
Both operations are **linear transformations** of the plane:
- scaling independently along $x$ and $y$ is the matrix $\mathrm{diag}(s_x,s_y)$;
- rotation about the origin is an orthogonal matrix.
A linear map fixes the origin and maps a circle centred at the origin to an **ellipse centred at the origin** (possibly with its axes tilted by the rotation).
So the shape that can be obtained must be an ellipse centred at O. In the options:
- one candidate is centred at O and is an ellipse; this is the one reachable. ✓
- another is shifted away from O, so it needs a translation, which is not allowed;
- the others are not ellipses (circles scaled and rotated never get corners or straight sides).
Answer **A**.
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