Q. 124.0( 2 Votes )
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
Answer :
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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