In a A B C , if | array lll 1 & a & b 1 & c & a 1 & b & c array |=0 , then ^2 A+ ^2 B+ ^2 C is equal to

Step-by-step solution

We have, aligned | array lll 1 & a & b 1 & c & a 1 & b & c array | & =0 | array ccc 1 & a & b 0 & c-a & a-b 0 & b-a & c-b array | & =0 aligned [applying R₂ R₂-R₁ and R₃ R₃-R₁ ] array ll & (c-a)(c-b)+(a-b)^2=0 & a^2+b^2+c^2-a b-b c-c a=0 & 2 a^2+2 b^2+2 c^2-2 a b-2 b c-2 c a=0 & (a-b)^2+(b-c)^2+(c-a)^2=0 & a=b=c array A B C is an equilateral triangle. A=B=C= 3 ^2 A+ ^2 B+ ^2 C=3 ^2 3 = 9 4