In ABC , if A B C = 1 5 , then A B + B C + C A =

Step-by-step solution

Given :- A B C= 1 5 A+B+C= A+B= -C aligned & (A+B)= ( -C) & A B- A B=- C & A B+ C= A B aligned 1+ C A B = A B or A B=1+ C A B ...(i) Similarly, we have, B C=1+ A B C ...(ii) C A=1+ B C A ...(iii) Adding eqn. (i), (ii) and (iii), we get A B+ B C+ C A=3+ C A B + A B C + B C A =3+ ^2 C+ ^2 A+ ^2 B A B C =3+5 ( ^2 C+ ^2 A+ ^2 B ) ...(iv) On A B C , ^2 A+ ^2 B+ ^2 C=1-2 A B C ...(v) Using eqn. (v) in eqn. (iv), we get A B+ B C+ C A=3+5(1-2 A B C)=3+5 (1-2 1 5 )=3+5 3 5 =3+3 A B+ B C+ C A=6 Option (3) is correct.