Consider the following linear equations in x and y : a x + b y + c = 0 , b x + cy + a = 0 , c x + a y + b = 0 . Match the condition in Column I with the statement in Column II. List-I List-II (A) a +

Question

Consider the following linear equations in $x$ and $y$ : $a x + b y + c = 0$ , $b x + cy + a = 0$ , $c x + a y + b = 0$ . Match the condition in Column I with the statement in Column II.

List-I	List-II
(A) $a + b + c = 0$ and $a^{2} + b^{2} + c^{2} = ab + b c + c a$	(p) Lines are identical
(B) $a + b + c = 0$ and $a^{2} + b^{2} + c^{2} \neq = ab + b c + c a$	(q) Lines represent the whole of the $x y$ -plane
(C) $a + b + c \neq = 0$ and $a^{2} + b^{2} + c^{2} = ab + b c + c a$	(r) Lines are different and passing through a fixed point
(D) $a + b + c \neq = 0$ and $a^{2} + b^{2} + c^{2} \neq = ab + b c + c a$	(s) Lines are the sides of a triangle

TestHub · Accepted Answer

(A) a 2 + b 2 + c 2 = ab + b c + c a forces a = b = c ; with a + b + c = 0 all coefficients vanish, so every ( x , y ) satisfies all three equations: the whole plane. (B) a + b + c = 0 makes ( 1 , 1 ) satisfy all three lines; with a , b , c not all equal the lines are distinct: concurrent at the fixed point (1, 1). (C) a = b = c  = 0 : all three equations reduce to x + y + 1 = 0 : identical lines. (D) Neither degeneracy: the three lines form a triangle.

Mathematics - Straight Lines Question with Solution | TestHub

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