If b+c/a=c+a/b=a+b/cand a+b+c is not equal to 0 then prove that a^2+b^2+c^2=ab+bc+ca
Answers
Answer:
Step-by-step explanation:
here , a ² + b² + c² – ab – bc – ca = 0
to prove a = b= c
solution :
by , multiplying both sides with 2, we get the result.
2( a ²+ b² + c² – ab – bc – ca) = 0
⇒ 2a² + 2b²+ 2c² – 2ab – 2bc – 2ca = 0
⇒ (a² – 2ab + b²) + (b² – 2bc + c²) + (c² – 2ca + a²) = 0
⇒ (a –b)² + (b – c)² + (c – a)² = 0
hence, we can say ,(a - b)² = (b - c)² = (c - a)² = 0
therefore,
we can say, (a - b)² = 0 ---------- (1)
(b - c)² = 0 ---------- (2)
(c - a)² = 0 ---------- (3)
therefore, by Simplifying Equ. (1), we get
(a - b)² = 0
now Taking the Square Root on both sides, we have
a - b = 0
a = b ---------- (4)
similarly, Simplifying Equ. (2), we get, (b - c)² = 0
Taking Square Root on both sides, we have
b - c = 0
b = c ---------- (5)
again, Simplifying Equ. (3), we have, (c - a)² = 0
Taking Square Root on both sides, we have
c - a = 0
c = a ---------- (6)
now, From Equation No. (4), (5) & (6) , it is proved that
a = b = c
Hence Proved.....
HOPE IT HELPS.........