(c) a, b, c are in H.P. only for all m As the points A(a, ma), B[b, (m + 1)b] and C[c, (m + 2)c] are collinear. Area of Δ ABC should be equal to zero.⇒ \(rac{1}{2}\)[x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2)] = 0⇒ \(rac{1}{2}\)[a {(m + 1) b – (m + 2) c} + b {(m + 2) c – ma} + c {ma – (m + 1)b}] = 0 ⇒ mab + ab – mac – 2ac + mbc + 2bc – mab + mac – mbc – bc = 0 ⇒ ab – 2ac + 2bc – bc = 0 ⇒ ab + bc = 2ac ⇒ b = \(rac{2ac}{a+c}\)∴ a, b, c are harmonic progression (H.P.) for all m.