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? A. An ellipse with its centre at the origin and its major axis along a diagonal (not aligned with x or y axis) B. An ellipse with its centre not at the origin C. A circle with its centre not at the origin D. An ellipse with its centre at the origin and its major axis along the x or y axis

GATE 2023 · General Aptitude · Image Rotation · medium

Answer: Option A — an ellipse centred at the origin with its major axis along a diagonal — can be achieved by first scaling the circle independently along x and y (producing an axis-aligned ellipse) and then rotating about the origin (tilting the major axis to a diagonal direction).

  1. Apply centre-preservation constraint: Both operations fix the origin. Therefore any figure produced must have its centre at the origin. Options B (ellipse with centre not at origin) and C (circle with centre not at origin) are immediately eliminated.
  2. Check Option D: axis-aligned ellipse centred at origin: This is achievable by Operation 1 alone with unequal scale factors. However, the question asks which can be achieved by a combination of the two operations. Option D is possible but note it only uses one operation and the question is looking for what new figures both operations together can produce.
  3. Check Option A: diagonal ellipse centred at origin: Start with the unit circle. Apply Operation 1 with sx != sy to get an axis-aligned ellipse x^2/sx^2 + y^2/sy^2 = 1. Then apply Operation 2 (rotate by, say, 45 degrees) to tilt the major axis to the diagonal. The resulting figure is an ellipse centred at the origin with its major axis at 45 degrees — precisely the figure in Option A.
  4. Confirm Option A is the answer: Option A (diagonal ellipse centred at origin) requires both operations: scaling to create the ellipse and rotation to tilt it. It satisfies centre-at-origin. Options B and C are impossible (centre displaced). Option D uses only one operation. The question asks which CAN be achieved — Option A is the figure that specifically requires the combination and is the intended correct answer.