If $a \neq b \neq c,$ write the condition for which the equations $(b - c) x + (c - a)$ $y + (a - b) = 0$ and $(b^{3} - c^{3}) x + (c^{3} - a^{3}) y + (a^{3} - b^{3} = 0$ represent the same line.

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Solution

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

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

represint the same line if

$\frac{b^{3} - c^{3}}{b - c} = \frac{c^{3} - a^{3}}{c - a} = \frac{a^{3} - b^{3}}{1 - b}$

$\Rightarrow b^{2} + c^{2} + b c = c^{2} + a^{2} + a c = a^{2} + b^{2} + a b$

If $b^{2} + c^{2} + b c = c^{2} + a^{2} + a c$

$\Rightarrow b^{2} - a^{2} = a c - b c$

$\Rightarrow (b - a) (b + a) = a c - b c$

$\Rightarrow b + a = - c$

$\Rightarrow a + b + c = 0$

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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

> 0

\frac{a^{3}}{b^{3}} + \frac{b^{3}}{c^{3}} + \frac{c^{3}}{a^{3}} \geq 3

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