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Given:

The equation (b – c)x + (c – a) y + (a – b) = 0 and (b³ – c³)x + (c³ – a³)y + (a³ – b³) = 0

To find:

The condition for which the equation (b – c)x + (c – a) y + (a – b) = 0 and (b³ – c³)x + (c³ – a³)y + (a³ – b³) = 0 represent the same line

Explanation:

The given lines are

(b − c)x + (c − a)y + (a − b) = 0 … (1)

(b³ − c³)x + (c³ − a³)y + (a³ − b³) = 0 … (2)

The lines (1) and (2) will represent the same lines if

⇒

⇒

∵ (a ≠ b ≠ c)

⇒ b² + bc + c² = c² + ac + a² and c² + ac + a² = a² + ab + b²

⇒ (a – b) (a + b + c) = 0 and (b – c) (b + c + a) = 0

⇒ a + b + c = 0 ∵(a ≠ b ≠ c)

Hence, the given lines will represent the same lines if a + b + c = 0.