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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.

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