Question

Consider the two curves $C_1:y^2=4x,C_2:x^2+y^2-6x+1=0.$ 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

For points of intersection of circle and parabola, put $y^2=4x$ in the equation of the circle, we get

$x^2+4x-6x+1=0or(x-1{)}^2=0$ i.e., they intersect at x = 1.
After substituting in the equation of parabola, we get
$y=2,-2$
Thus they touch at the above two points.