Find $\frac{x^2}{y^2}$ + $\frac{y^2}{x^2}$

Question

If $\frac{x}{y}$ + $\frac{y}{x}$ = 3 Find $\frac{x^2}{y^2}$ + $\frac{y^2}{x^2}$

Any Ideas on how to begin ?

Answer 1

\begin{align*} \frac{x}{y} + \frac{y}{x} = 3 &\Longrightarrow \left(\frac{x}{y} + \frac{y}{x}\right)^2 = 9 \\ &\Longrightarrow \frac{x^2}{y^2} + \frac{y^2}{x^2} = 9-2=7 \end{align*}

Answer 2

Start by squaring both sides of $\frac{x}{y}+\frac{y}{x}=3$. What happens to the cross terms?

Answer 3

Hint: Look at:

$$\left(\frac{x}{y} + \frac{y}{x}\right)^2=3^2=9$$

Why to look at this? We know if we expand we will have a $(\frac{x}{y})^2=\frac{x^2}{y^2}$ term and a $(\frac{y}{x})^2=\frac{y^2}{x^2}$ term so this might be worth a try.