Question

if a^3+b^3+c^3=3abc then which of the following can be values of (a+b)/c+(b+c)/a+(c+a)/b?​

Answer 1

If “a3+b3+c3=3abc then find the value of (a+b+c) where a≠b≠c”?

Value of (a+b+c) is zero.

Why?

we know,

a³ + b³ + c³ - 3abc = (a + b + c )(a² + b² + c² -ab - ac -bc)

Now it is given that : a³ + b³ + c³ = 3abc

So,

a³ + b³ + c³ - 3abc = 0

(a + b + c )(a² + b² + c² -ab - ac -bc) = 0

this means either

(a² + b² + c² -ab - ac -bc) = 0 or (a + b + c ) = 0

(a² + b² + c² -ab - ac -bc) = 0

cannot be zero because:

2a² + 2b² + 2c² -2ab - 2ac - 2bc = 0

a² + b² -2ab + a² +b² +2c² - 2ac -2bc = 0

(a-b)² + a² + c² -2ac + b² + c² -2bc = 0

(a-b)² + (a-c)² + (b-c)² = 0

(a-b)² , (a-c)² ,(b-c)² >=0

As a≠b≠c , The given value cannot be zero.

This means (a + b + c ) has to be zero.