X and Y are two variables whose values at any time are related to each other as shown in Fig. (i). X is known to vary periodically with reference to time as shown in Fig. (ii). Which of the following curves depicts correctly the dependence of Y on time?

The X–Y plot in Fig. (i) represents a circular (Lissajous) relation between the two variables, which implies parametric equations x = R cos(ωt), y = R sin(ωt) (or with a sign change) when X varies sinusoidally in time. Thus Y must also vary sinusoidally with the same frequency but be in quadrature (±90° phase shift) with X, producing a sine wave shifted by one quarter period relative to X. Hence the correct Y(t) curve is the sinusoid that is phase–shifted by π/2 with respect to X(t), as shown in option III. This follows from the graphical-function idea and standard Lissajous/phase relations for circular plots [2].

Sources

1. [1] Microeconomics (NCERT class XII 2025 ed.) > Chapter 2: Theory of Consumer Behaviour > EXAMPLE 2 > p. 22
2. [2] https://en.wikipedia.org/wiki/Lissajous_curve