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

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

c = a

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

a = b

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

a + b + c = 0

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

Open in App

Solution

The correct options are
A $b = c$
B $c = a$
C $a = b$
D $a + b + c = 0$
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 $a^{3} - b^{3} = k (a - b)$

$\Rightarrow b - c = 0 o r b^{2} + c^{2} + b c = k$

$c - a = 0 o r c^{2} + a^{2} + c a = k$

and $a - b = 0 o r a^{2} + b^{2} + a b = k$

That is , $b = c$ or $c = a$ or $a = b$

Next $b^{2} + c^{2} + b c = c^{2} + a^{2} + c a \Rightarrow b^{2} - a^{2} = c (a - b)$

Hence, $a = b$ or $a + b + c = 0.$

Suggest Corrections

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

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}

If $a + b + c = 0$ , then the value of $(a + b - c)^{3} + (b + c - a)^{3} + (c + a - b)^{3}$ is:

Join BYJU'S Learning Program