Consider the two curves $C_1: y^2=4 x$ $C_2: x^2+y^2-6 x+1=0$, then
- $C_1$ and $C_2$ touch each other only at one point
- $C_1$ and $C_2$ touch each other exactly at two points
- $C_1$ and $C_2$ intersect (but do not touch) at exactly two points
- $C_1$ and $C_2$ neither intersect nor touch each other
Solution
For the points of intersection of the two given curves $ C_1: y^2=4 x \text { and } C_2: x^2+y^2-6 x+1=0 \text {, } $ we have $ \begin{aligned} & x^2+4 x-6 x+1=0 \\ & \Rightarrow \quad x^2-2 x+1=0 \\ & \Rightarrow \quad(x-1)^2=0 \\ & \Rightarrow \quad x=1,1 \\ & \Rightarrow \quad y=2,-2 \\ & \end{aligned} $ Thus, the given curves touch each other exactly at two points $(1,2)$ and $(1,-2)$
Asked in: JEE Advanced 2008 (Paper 1)