If a b c, write 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

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.


12