Question

If $x+y+2=0$, then, the value of $3\left[\frac{x^2}{y z}+\frac{y^2}{z x}+\frac{z^2}{x y}\right]$, is: ____

Answer 1

We have,
$ x+y+2=0 \quad \text { (Given) } $...(1)
$ \text { Now, } 3\left[\frac{x^2}{y z}+\frac{y^2}{z x}+\frac{z^2}{x y}\right]$
$ =3\left[\frac{x^2}{y z} \times \frac{x}{x}+\frac{y^2}{z x} \times \frac{y}{y}+\frac{z^2}{x y} \times \frac{z}{z}\right] $
$ =3\left[\frac{x^3}{x y z}+\frac{y^3}{x y z}+\frac{z^3}{x y z}\right]$
$ =3\left[\frac{x^3+y^3+z^3}{x y z}\right]$....(2)
Also,
We know that
$ x^3+y^3+z^3-3 x y z $
$ =(x+y+z)\left(x^2+y^2+z^2-x y-y z-z x\right)$
$ \Rightarrow x^3+y^3+z^3-3 x y z=0 \text { (From equation (1)) }$
$ \Rightarrow x^3+y^3+z^3=3 x y z $...(3)
Putting value of equation (3) in equation (2)
$3\left[\frac{3 x y z}{x y z}\right]=3 \times 3=9$