x/y=1/3then value of x^2+y^2/x^2-y^2
You asked:
Given \(\frac{x}{y} = \frac{1}{3}\) evaluate the expression \({x}^{2} + \frac{{y}^{2}}{{x}^{2}} - {y}^{2}\).
MathBot Answer:
\[x^{2} - y^{2} + \frac{y^{2}}{x^{2}} = 9 - \frac{8 y^{2}}{9}\]
\[\frac{x}{y} = \frac{1}{3}\]Solve for $x$: \[x = \frac{y}{3}\]Substitute $x$ into the given expression: $$\begin{aligned}x^{2} - y^{2} + \frac{y^{2}}{x^{2}} &= - y^{2} + \frac{y^{2}}{\left(\frac{y}{3}\right)^{2}} + \left(\frac{y}{3}\right)^{2} \\ x^{2} - y^{2} + \frac{y^{2}}{x^{2}} &= 9 - \frac{8 y^{2}}{9}\end{aligned}$$