Question

If the roots of (a – b)x² + (b – c)x + (c – a) = 0 are real and equal, then prove that b, a, c are in arithmetic progression

Answer 1

(a – b)x² + (b – c)x + (c – a) = 0

Here a = (a – b); b = b – c; c = c – a

Since the equation has real and equal roots ∆ = 0

∴ b² – 4ac = 0

(b – c)² – 4(a – b)(c – a) = 0

b² + c² – 2bc – 4 (ac – a² – bc + ab) = 0

b² + c² – 2bc – 4ac + 4a² + 4bc – 4ab = 0

b² + c² + 2bc – 4a (b + c) + 4a² = 0

(b + c)² – 4a (b + c) + 4a² = 0

[(b+c) – 2a]² = 0  ...[using a² – 2ab + b² = (a – b)²]

b + c – 2a = 0

b + c = 2a

b + c = a + a

c – a = a – b  ...(t₂ – t₁ = t₃ – t₂)

b, a, c are in A.P.