Question

If a2 + b2 +4 c2 = ab +2bc +2ca ,prove that a = b = 2c .

Answer 1

Answer:

hey mate here is your answer

Step-by-step explanation:

a^2+ b^2 + c^2 = ab + bc + ac

or 2a^2+2b^2+2c^2 = 2ab + 2bc + 2ac

or (a^2-2ab+b^2)+(b^2-2bc+c)^2+(c^2-2ac+a^2) =0

or( a-b)^2+( b-c)^2 +(c-a)^2 =0

now sum of squares will be zero only if all the squares are

equal to zero

so ( a-b)^2=0 or a= b

(b-c)^2 =0 or b = c

(c-a )^2 =0 or c = a

so a = b = c proved

Answer 2

Consider, a2 + b2 + c2 – ab – bc – ca = 0 Multiply both sides with 2, we get 2( a2 + 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 Since the sum of square is zero then each term should be zero ⇒ (a –b)2 = 0,  (b – c)2 = 0, (c – a)2 = 0 ⇒ (a –b) = 0,  (b – c) = 0, (c – a) = 0 ⇒ a = b,  b = c, c = a ∴ a = b = c