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Question
Prove that the curves xy = 4 and x2 + y2 = 8 touch each other ?
Solution
\[\text { Given }: \]
\[xy = 4 . . . . . \left( 1 \right)\]
\[ x^2 + y^2 = 8 . . . . . \left( 2 \right)\]
\[\text { From } \left( 1 \right), \text { we get }\]
\[x = \frac{4}{y}\]
\[\text { Substituting } x = \frac{4}{y} \text { in }\left( 2 \right), \text { we get }\]
\[ \left( \frac{4}{y} \right)^2 + y^2 = 8\]
\[ \Rightarrow \frac{16}{y^2} + y^2 = 8\]
\[ \Rightarrow 16 + y^4 = 8 y^2 \]
\[ \Rightarrow y^4 - 8 y^2 + 16 = 0\]
\[ \Rightarrow \left( y^2 - 4 \right)^2 = 0\]
\[ \Rightarrow y^2 - 4 = 0\]
\[ \Rightarrow y^2 = 4\]
\[ \Rightarrow y = \pm 2\]
\[\text { Substituting }y = \pm 2, \text { we get }\]
\[x = \pm 2\]
\[\text { So, the given curves touch each other at two points } \left( 2, 2 \right) \text { and } \left( - 2, - 2 \right) .\]
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