If $a+b+c=0$, then the value of $\frac{a^4+b^4+c^4}{a^2{b}^2+b^2{c}^2+c^2{a}^2}$ will be:

Algebra

Linear equations in two variables: If $a, b,$ and $r$ are real numbers (and if $a$ and $b$ are not both equal to $0$) then $ax + by = r$ is called a linear equation in two variables.

The pair of linear equations represented by lines $1x a+ b1y + c1 = 0$ and $a2x + b2y + c2 = 0$

If $\frac{a1}{a2}!=\frac{b1}{b2}$ then the pair of linear equations has exactly one solution.

If $\frac{a1}{a2}=\frac{b1}{b2}=\frac{c1}{c2}$ then the pair of linear equations has infinitely many solutions.

If $\frac{a1}{a2}=\frac{b1}{b2}=\frac{c1}{c2}$  then the pair of linear equations has no solution.

Algebraic Identities:

1. $a^2 – b^2 = \left(a – b\right)\left(a + b\right)$ 

2. $\left(a + b\right)2 = a^2 + 2ab + b^2$

3. $a^2 + b^2= {\left(a – b\right)}^2+ 2ab$

5. ${\left(a + b + c\right)}^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc$

7. $a^3 + b^3 + c^3 – 3abc =\left(a + b+ c\right)\left(a^2 + b^2 + c^2 – ab – ac – bc\right)$

9. ${\left(a + b\right)}^3 = a^3 + 3a^{2}b + 3a{b}^2 + b3 ; {\left(a + b\right)}^3 = a^3 + b^3 + 3ab(a + b)$

11. $a^3 – b^3 =\left(a – b\right)\left(a^2 + ab + b^2\right)$

13. ${\left(a + b\right)}^3 = a^3 + 3a^{2}b + 3a{b}^2 + b^3$

15. ${\left(a + b\right)}^4 = a^4 + 4a^{3}b + 6a^2{b}^2 + 4a{b}^3 + b^4$

17. $a^4 – b^4 = \left(a – b\right)\left(a + b\right)\left(a^2 + b^2\right)$

19. If$a + b + c = 0$; then

20. $ab\left(a + b\right) + bc\left(c + b\right) + ca\left(c + a\right) =\left(a + b\right)\left(b + c \right)\left(c + a\right)$

22. $a^2\left(b – c\right) + b^2 \left(c – a\right) + c^2 \left(a – b\right) = \left(a – b\right)\left(b –c\right)\left(c – a\right)$

24. If $n$ is even $\left(n = 2k\right)$, $a^n + b^n =\left(a + b\right)\left(an - 1 – an - 2b + \dots + bn - 2a – bn - 1\right)$

25. If $n$ is odd $\left(n = 2k + 1\right)$,

26.

$a^n + b^n = \left(a + b\right)\left(a^n - 1 – a^n - 2b + \dots – b^{n }- 2a + b^n - 1\right)$

27.  ${(a + b + c + \dots )}^2 = a^2 + b^2 + c^2 + \dots + 2\left(ab + ac + bc + \dots .\right)$

Let us consider the equation $A{x}^2 + Bx + C = 0, A{y}^2 + By + C = 0$

Sum of its roots $=$$-$$BA$ and product of its roots $=$$CA$.