Question

If $a,b,c$ are real, ${a^3} + {b^3} + {c^3} - 3abc = 0$ and $a + b + c \ne 0$ , then the relation between $a,b,c$ will be:
$\left( a \right){\text{ a + b = c}}$
$\left( b \right){\text{ a + c = b}}$
$\left( c \right){\text{ a = b = c}}$
$\left( d \right){\text{ b + c = a}}$

Answer 1

Hint:
In this question, we can see that $a,b,c$ are real and $a + b + c \ne 0$ so by using the formula ${a^3} + {b^3} + {c^3} - 3abc = (a + b + c)({a^2} + {b^2} + {c^2} - ab - bc - ca)$ and then we will compare the left and right-hand side of the equation, then in this way we will justify the relation.
Formula used:
${a^3} + {b^3} + {c^3} - 3abc = (a + b + c)({a^2} + {b^2} + {c^2} - ab - bc - ca)$
And another formula is
${\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab$
Here, the variables are used just for reference.

Complete step by step solution:
So in the question, it is given that they $a,b,c$ are real and $a + b + c \ne 0$
Since from the formula, we know that ${a^3} + {b^3} + {c^3} - 3abc = (a + b + c)({a^2} + {b^2} + {c^2} - ab - bc - ca)$
So, ${a^3} + {b^3} + {c^3} - 3abc = 0$ can also be written as
$ \Rightarrow (a + b + c)({a^2} + {b^2} + {c^2} - ab - bc - ca) = 0$
And on solving it with the help of information given in the question, the equation will be
\[ \Rightarrow ({a^2} + {b^2} + {c^2} - ab - bc - ca) = 0\]
Now on multiplying by $2$ on both the sides of the equation, we get
\[ \Rightarrow 2 \times ({a^2} + {b^2} + {c^2} - ab - bc - ca) = 0 \times 2\]
So on solving the above equation and expanding the above equation, we get
$ \Rightarrow \left( {{a^2} + {b^2} - 2ab} \right) + \left( {{a^2} + {c^2} - 2ac} \right) + \left( {{b^2} + {c^2} - 2bc} \right) = 0$
And we know the above equation follows the pattern of the formula ${\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab$ , we get the equation as
$ \Rightarrow {\left( {a - b} \right)^2} + {(a - c)^2} + {(b - c)^2} = 0$
Since the left side of the equation is equal to the right side of the equation, then we can say that
$ \Rightarrow a - b = 0,{\text{ }}a - c = 0,{\text{ }}b - c = 0$
So from this $a = b = c$ and hence our equation justified.

Therefore, the option $\left( c \right)$ is correct.

Note:
In this type of question the main important thing is the formula, so we should memorize such formulas as it is always used. And also we should know that the real numbers can be in any form like in fractions or it may be negative or positive. For solving the problem related to proof or justification, we should always move forward with the equations, given in the question either by taking the whole equation or by taking only LHS or RHS of it.