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An opaque cylinder (shown below) is suspended in the path of a parallel beam of light, such that its shadow is cast on a screen oriented perpendicular to the direction of the light beam. The cylinder can be reoriented in any direction within the light beam. Under these conditions, which one of the shadows P, Q, R, and S is NOT possible? The four candidate shadows are: P. A circle Q. A rectangle (taller than wide) R. A rectangle (wider, with rounded ends / ellipse-like) S. A triangle A. P B. Q C. R D. S

GATE 2023 · General Aptitude · Patterns In Three Dimensions · medium

Answer: Shadow S (a triangle) is NOT possible because a cylinder's circular end caps always project as a circle or ellipse, never a sharp point, so the shadow can never have the converging straight sides of a triangle.

  1. Shadow when axis is parallel to light beam (theta = 0 deg): When the light beam travels exactly along the cylinder axis, the shadow is the projection of the circular end cap: a circle of radius r. This matches shadow P.
  2. Shadow when axis is perpendicular to light beam (theta = 90 deg): When the axis is fully perpendicular, the shadow is a rectangle. The width is the diameter 2r and the height is the axial length L. Shadows Q (tall rectangle) and R (wide rectangle) are both achievable by choosing different axis orientations relative to the up direction on the screen.
  3. Shadow at intermediate angle (0 < theta < 90 deg): At any intermediate tilt, each circular end cap projects as an ellipse (since it is viewed at an angle). The shadow becomes a 'stadium' or 'discorectangle' shape: a rectangle with rounded (elliptical) ends. This is still a version of R.
  4. Assess triangle shadow S: A triangle requires a pointed apex. The cylinder's end caps are always circular; their projection is always a circle or an ellipse, never a point. The side of the cylinder projects as two parallel straight lines. Therefore the shadow boundary always has at least one pair of smooth curved ends and can never form a triangle with straight converging sides. Shadow S is NOT possible.