Let α be a root of the equation (a-c) x2+(b-a) x+(c-b)=0 where a , b , c are distinct real numbers such that the matrix [α2 α 1 1 1 1 a b c] is singular. Then, the value of ((a-c)2/(b-a)(c-b))+((b-a)2/(a-c)(c-b))+((c-b)2/(a-c)(b-a)) is

Question

Let α be a root of the equation (a-c) x2+(b-a) x+(c-b)=0 where a , b , c are distinct real numbers such that the matrix [α2 α 1 1 1 1 a b c] is singular. Then, the value of ((a-c)2/(b-a)(c-b))+((b-a)2/(a-c)(c-b))+((c-b)2/(a-c)(b-a)) is (A) 6 (B) 3 (C) 9 (D) 12

Tardigrade Admin · Accepted Answer

Δ=0=|α2 α 1 1 1 1 a b c| ⇒ α2(c-b)-α(c-a)+(b-a)=0 It is singular when α=1 ((a-c)2/(b-a)(c-b))+((b-a)2/(a-c)(c-b))+((c-b)2/(a-c)(b-a)) ((a-b)3+(b-c)3+(c-a)3/(a-b)(b-c)(c-a)) =3 ((a-b)(b-c)(c-a)/(a-b)(b-c)(c-a))=3

Q. Let α be a root of the equation (a−c)x2+(b−a)x+(c−b)=0 where a,b,c are distinct real numbers such that the matrix ⎣⎡​α21a​α1b​11c​⎦⎤​ is singular. Then, the value of (b−a)(c−b)(a−c)2​+(a−c)(c−b)(b−a)2​+(a−c)(b−a)(c−b)2​ is

6

3

9

12

Δ=0=∣∣​α21a​α1b​11c​∣∣​ ⇒α2(c−b)−α(c−a)+(b−a)=0 It is singular when α=1 (b−a)(c−b)(a−c)2​+(a−c)(c−b)(b−a)2​+(a−c)(b−a)(c−b)2​ (a−b)(b−c)(c−a)(a−b)3+(b−c)3+(c−a)3​ =3(a−b)(b−c)(c−a)(a−b)(b−c)(c−a)​=3