Question

Prove that the curves xy = 4 and x² + y² = 8 touch each other. [NCERT EXEMPLAR]

Answer 1

$$\begin{gathered} \mathrm{Given}: \\ xy=4.....\left(1\right) \\ {x}^2+y^2=8.....\left(2\right) \\ \mathrm{From}\left(1\right),\mathrm{we}\mathrm{get} \\ x=\frac{4}{y} \\ \mathrm{Substituting}x=\frac{4}{y}\mathrm{in}\left(2\right),\mathrm{we}\mathrm{get} \\ {\left(\frac{4}{y}\right)}^2+y^2=8 \\ \Rightarrow \frac{16}{y^2}+y^2=8 \\ \Rightarrow 16+y^4=8y^2 \\ \Rightarrow y^4-8y^2+16=0 \\ \Rightarrow {\left(y^2-4\right)}^2=0 \\ \Rightarrow y^2-4=0 \\ \Rightarrow y^2=4 \\ \Rightarrow y=\pm 2 \\ \mathrm{Substituting}y=\pm 2,\mathrm{we}\mathrm{get} \\ x=\pm 2 \\ \mathrm{So},\mathrm{the}\mathrm{given}\mathrm{curves}\mathrm{touch}\mathrm{each}\mathrm{other}\mathrm{at}\mathrm{two}\mathrm{points}\left(2,2\right)\mathrm{and}\left(-2,-2\right). \end{gathered}$$