For real a,b,c, consider ax+by+c=0, bx+cy+a=0 and cx+ay+b=0. Match each condition to its geometric description. List I List II (A) a+b+c=0 and a²+b²+c²=ab+bc+ca (P) Identical lines (B) a+b+c=0 and a²+

Question

For real a,b,c, consider ax+by+c=0, bx+cy+a=0 and cx+ay+b=0. Match each condition to its geometric description.

List I	List II
(A) a+b+c=0 and a²+b²+c²=ab+bc+ca	(P) Identical lines
(B) a+b+c=0 and a²+b²+c²≠ab+bc+ca	(Q) Every point of the xy-plane satisfies the equations
(C) a+b+c≠0 and a²+b²+c²=ab+bc+ca	(R) Three different lines through a common fixed point
(D) a+b+c≠0, a²+b²+c²≠ab+bc+ca, and all three line pairs are nonparallel	(S) Sides of a triangle

TestHub · Accepted Answer

a 2 + b 2 + c 2 − ( ab + b c + c a ) = 2 1 ​ [ ( a − b ) 2 + ( b − c ) 2 + ( c − a ) 2 ] . In (A), all coefficients vanish. In (B), all three lines pass through ( 1 , 1 ) and are distinct. In (C), a = b = c  = 0 , so the lines coincide. In (D), the determinant of the three line coefficients is ( a + b + c ) [ a 2 + b 2 + c 2 − ab − b c − c a ]  = 0 , precluding concurrency; the added pairwise nonparallel condition ensures a triangle.

Mathematics - Straight Lines Question with Solution | TestHub

Options:

Doubts & Discussion