Question : If $x + y + z = 0$ , then what is the value of $\frac{x^{2}}{(y z)} + \frac{y^{2}}{(x z)} + \frac{z^{2}}{(x y)}$ ?
Option 1: 1
Option 2: 0
Option 3: 2
Option 4: 3

Question

Question : If $\frac{(x + y)}{z} = 2$ , then what is the value of $[\frac{y}{(y - z)} + \frac{x}{(x - z)}] ?$
Option 1: $0$
Option 2: $1$
Option 3: $2$
Option 4: $- 1$

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Answer 1

Correct Answer: 3

Solution : Given: $x + y + z = 0$
Cubing both sides, we get,
$\Rightarrow (x + y + z)^{3} = 0$
$\Rightarrow x^{3} + y^{3} + z^{3} + 3 (x + y) (y + z) (z + x) = 0$
$\Rightarrow x^{3} + y^{3} + z^{3} = - 3 (x + y) (y + z) (z + x)$ ......(1)
From the given equation ( $x + y + z = 0$ ), we can get,
$x + y = - z,$ $y + z = - x$ and $z + x = - y$
Putting in (1), we get,
$x^{3} + y^{3} + z^{3} = - 3 (- z) (- x) (- y) = 3 x y z$
Consider, $\frac{x^{2}}{(y z)} + \frac{y^{2}}{(x z)} + \frac{z^{2}}{(x y)}$
$= \frac{x^{2} (x z) (x y) + y^{2} (y z) (x y) + z^{2} (y z) (x z)}{(y z) (x z) (x y)}$
$= \frac{x^{4} y z + y^{4} z x + z^{4} x y}{x^{2} y^{2} z^{2}}$
Taking $x y z$ as common, we get,
$= \frac{x y z (x^{3} + y^{3} + z^{3})}{x^{2} y^{2} z^{2}}$
$= \frac{x^{3} + y^{3} + z^{3}}{x y z}$
$= \frac{3 x y z}{x y z}$
$= 3$
Hence, the correct answer is 3.

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Question : If x+y+z=0, then what is the value of x2(yz)+y2(xz)+z2(xy)?Option 1: 1Option 2: 0Option 3: 2Option 4: 3

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Question : If $x + y + z = 0$ , then what is the value of $\frac{x^{2}}{(y z)} + \frac{y^{2}}{(x z)} + \frac{z^{2}}{(x y)}$ ?
Option 1: 1
Option 2: 0
Option 3: 2
Option 4: 3