Consider the two curves C₁ : y² = 4xC₂ : x² + y² – 6x + 1 = 0Then,

The correct answer is: C₁ and C₂ touch each other exactly at two points

To find the correct option.

Solving the two equations, we get  $x=1$ 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 C1​ is $y(2)=2(x+1)$
$y=x+1$
and Equation of the tangent at $(1,2)$ to C2​ is
$x.1+y(2)-3(x+1)+1=0$

y=x+1

Showing that C1​ and C2​ 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.