Question

Prove that the curves y² = 4x and x² + y² $-$ 6x + 1 = 0 touch each other at the point (1, 2). [NCERT EXEMPLAR]

Answer 1

$$\begin{gathered} \mathrm{Given}: \\ {y}^2=4x.....\left(1\right)\mathrm{and} \\ {x}^2+y^2-6x+1=0.....\left(2\right) \\ \mathrm{From}\left(1\right)\mathrm{and}\left(2\right),\mathrm{we}\mathrm{get} \\ {x}^2+4x-6x+1=0 \\ \Rightarrow x^2-2x+1=0 \\ \Rightarrow {\left(x-1\right)}^2=0 \\ \Rightarrow x-1=0 \\ \Rightarrow x=1 \\ \mathrm{Substititing}x=1\mathrm{in}\left(1\right),\mathrm{we}\mathrm{get} \\ {\mathrm{y}}^2=4 \\ \Rightarrow y=\pm 2 \\ \mathrm{So},\mathrm{the}\mathrm{two}\mathrm{given}\mathrm{curves}\mathrm{touch}\mathrm{each}\mathrm{other}\mathrm{at}\mathrm{two}\mathrm{points}\left(1,2\right)\mathrm{and}\left(1,-2\right). \end{gathered}$$