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Question
Prove that the curves y2 = 4x and x2 + y2 - 6x + 1 = 0 touch each other at the point (1, 2) ?
Solution
\[\text { Given }: \]
\[ y^2 = 4x . . . . . \left( 1 \right) \text { and }\]
\[ x^2 + y^2 - 6x + 1 = 0 . . . . . \left( 2 \right)\]
\[\text { From} \left( 1 \right) and \left( 2 \right), \text { we get }\]
\[ x^2 + 4x - 6x + 1 = 0\]
\[ \Rightarrow x^2 - 2x + 1 = 0\]
\[ \Rightarrow \left( x - 1 \right)^2 = 0\]
\[ \Rightarrow x - 1 = 0\]
\[ \Rightarrow x = 1\]
\[\text { Substititing } x = 1 in \left( 1 \right), \text { we get }\]
\[ y^2 = 4\]
\[ \Rightarrow y = \pm 2\]
\[\text { So, the two given curves touch each other at two points} \left( 1, 2 \right) \text { and } \left( 1, - 2 \right) .\]
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