Question

If x²y² + k² = 2kxy, where k is constant, then prove that y varies inversely as x.

Answer 1

$$\begin{gathered} x^2{y}^2+k^2=2kxy \\ \Rightarrow x^2{y}^2+k^2-2kxy=0 \\ \Rightarrow {\left(xy-k\right)}^2=0 \\ \Rightarrow xy-k=0 \\ \Rightarrow xy=k \\ \Rightarrow y=\frac{k}{x} \\ \therefore y\alpha \frac{1}{x} \end{gathered}$$