Question

The value of $\frac{(\frac{x}{y}-\frac{y}{x})(\frac{y}{z}-\frac{z}{y})(\frac{z}{x}-\frac{x}{z})}{(\frac{1}{x^2}-\frac{1}{y^2})(\frac{1}{y^2}-\frac{1}{z^2})(\frac{1}{z^2}-\frac{1}{x^2})}$ is

• A. $-x^2{y}^2{z}^2$
• B. $x^2{y}^2{z}^2$
• C. 1
• D. xyz

Answer 1

The correct option is A $-x^2{y}^2{z}^2$

Consider the given expression.

$\Rightarrow \frac{(\frac{x}{y}-\frac{y}{x})(\frac{y}{z}-\frac{z}{y})(\frac{z}{x}-\frac{x}{z})}{(\frac{1}{x^2}-\frac{1}{y^2})(\frac{1}{y^2}-\frac{1}{z^2})(\frac{1}{z^2}-\frac{1}{x^2})}$

$\Rightarrow \frac{\left(\frac{x^2-y^2}{xy}\right)\left(\frac{y^2-z^2}{yz}\right)\left(\frac{z^2-x^2}{xz}\right)}{\left(\frac{y^2-x^2}{x^2{y}^2}\right)\left(\frac{z^2-y^2}{z^2{y}^2}\right)\left(\frac{x^2-z^2}{x^2{z}^2}\right)}$

$\Rightarrow \frac{\left(\frac{x^2-y^2}{xy}\right)\left(\frac{y^2-z^2}{yz}\right)\left(\frac{z^2-x^2}{xz}\right)}{-\left(\frac{x^2-y^2}{x^2{y}^2}\right)\left(\frac{y^2-z^2}{z^2{y}^2}\right)\left(\frac{z^2-x^2}{x^2{z}^2}\right)}$

$\Rightarrow -\left(\frac{x^2{y}^2}{xy}\right)\left(\frac{z^2{y}^2}{yz}\right)\left(\frac{x^2{z}^2}{xz}\right)$

$\Rightarrow -x^2{y}^2{z}^2$

Hence, this is the answer.