Question

If a ≠ b ≠ c, write the condition for which the equations (b − c) x + (c − a) y + (a − b) = 0 and (b³ − c³) x + (c³ − a³) y + (a³ − b³) = 0 represent the same line.

Answer 1

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

\[\frac{b - c}{b^3 - c^3} = \frac{c - a}{c^3 - a^3} = \frac{a - b}{a^3 - b^3}\]

\[ \Rightarrow \frac{b - c}{\left( b - c \right)\left( b^2 + bc + c^2 \right)} = \frac{c - a}{\left( c - a \right)\left( c^2 + ac + a^2 \right)} = \frac{a - b}{\left( a - b \right)\left( a^2 + ab + b^2 \right)}\]

\[ \Rightarrow \frac{1}{b^2 + bc + c^2} = \frac{1}{c^2 + ac + a^2} = \frac{1}{a^2 + ab + b^2} \left( \because a \neq b \neq c \right)\]

\[\Rightarrow b^2 + bc + c^2 = c^2 + ac + a^2 \text { and } c^2 + ac + a^2 = a^2 + ab + b^2 \]

\[ \Rightarrow \left( a - b \right)\left( a + b + c \right) = 0 \text { and } \left( b - c \right)\left( b + c + a \right) = 0\]

\[ \Rightarrow a + b + c = 0 \left( \because a \neq b \neq c \right)\]

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