Question

Consider the two curves $C_1:y^2=4x$, $C_2:x^2+y^2-6x+1$, then

• A. $C_1$ and $C_2$ touch each other only at one point.
• B. $C_1$ and $C_2$ touch each other exactly at two points
• C. $C_1$ and $C_2$ intersect (but do not touch) at exactly two points,
• D. $C_1$ and $C_2$ neither intersect nor touch each other.

Answer 1

The correct option is B $C_1$ and $C_2$ touch each other exactly at two points

Solving the two equations, we get
$x^2+4x-6x+1=0$
$\Rightarrow$ $x^2+2x+1=0$
$\Rightarrow$ $x=1$ and $y=\pm 2$
So the two curves meet at two points $(1,2)$ and $(1,-2).$
Equation of the tangent at $(1,2)$ to $C_1$ is $y\left(2\right)=2(x+1)$
$\Rightarrow y=x+1$
and Equation of the tangent at $(1,2)$ to $C_2$ is
$x.1+y\left(2\right)-3(x+1)+1=0\Rightarrow y=x+1$
Showing that $C_1$ and $C_2$ have a common tangent at the point $(1,2).$
Similarly they have a common tangent
$y=-(x+1)at(1,-2)$
Hence, the two curves touch each other exactly at two points.