The equations $(b - c) x + (c - a) y + a - b = 0$ and
$(b^{3} - c^{3}) x + (c^{3} - a^{3}) y + a^{3} - b^{3} = 0$ will represent the same line if

b = c

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c = a

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a = b

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a + b - c = 0

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Solution

The correct options are
A $b = c$
B $c = a$
C $a = b$
The two lines will be identical if there exists some real number $k$ such that
$b^{3} - c^{3} = k (b - c), c^{3} - a^{3} = k (c - a)$ and
$a^{3} - b^{3} = k (a - b)$
$\Rightarrow b - c = 0$ or $b^{2} + c^{2} + b c = k$
$c - a = 0$ or $c^{2} + a^{2} + c a = k$
$a - b = 0$ or $a^{2} + b^{2} + b c = k$
$\Rightarrow b = c$ or $c = a$ or $a = b$
or $a = b$ or $a + b + c = 0$

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Similar questions

(b - c) x + (c - a) y + (a - b) = 0

(b^{3} - c^{3}) x + (c^{3} - a^{3}) y + a^{3} - b^{3} = 0

(b - c) x + (c - a) y + (a - b) = 0

(b^{3} - c^{3}) x + (c^{3} - a^{3}) y + a^{3} - b^{3} = 0

a x + b y + c = 0, b x + c y + a = 0

c x + a y + b = 0

a^{3} + b^{3} + c^{3} = 3 a b c

a + b + c = 0

a + b + c = 0

(a + b - c)^{3} + (b + c - a)^{3} + (c + a - b)^{3}

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