Hyperbola

A rectangular hyperbola has equal transverse and conjugate axes, meaning a = b, which forces a fixed eccentricity regardless of the constant in xy = c^2. For any hyperbola e^2 = 1 + b^2/a^2, and setting b = a gives e^2 = 1 + 1 = 2, so e = sqrt(2). The form xy = 8 is the asymptotes-as-axes representation of a rectangular hyperbola, whose perpendicular asymptotes are the coordinate axes, a hallmark of the a = b case. Option 2 squares the eccentricity instead of taking the root. Option sqrt(3) corresponds to b^2/a^2 = 2, not the rectangular condition. Option 1 would describe a degenerate non-hyperbolic curve since e must exceed 1. This relies on the standard JEE Advanced rectangular-hyperbola property. Plausibility check: e = sqrt(2) ≈ 1.414 exceeds 1 as every hyperbola requires, and it is the smallest eccentricity consistent with equal axes, confirming the curve is a proper rectangular hyperbola.