(c) 1
Let each ratio = k and base = e
⇒ loge x = k(a2 + ab + b2)
⇒ (a – b) loge x = k (a – b) (a2 + ab + b2)
⇒ loge xa – b = k(a3 – b3) ⇒ xa – b = \(e^{k(a^3-b^3)}\)
Similarly, yb-c = \(e^{k(b^3-c^3)}\), zc-a = \(e^{k(c^3-a^3)}\)
∴ xa-b . yb-c . zc-a = \(e^{k(a^3-b^3)}\). \(e^{k(b^3-c^3)}\) . \(e^{k(c^3-a^3)}\)
= \(e^{k[a^3-b^3+b^3-c^3+c^3-a^3]}\) = e0 = 1.