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

Answer: (B)

$C_1$ and $C_2$ touch each other exactly at two points

Answer 2

Substitute $y^2 = 4x$ into the circle equation $x^2 + y^2 - 6x + 1 = 0$. This yields $x^2 - 2x + 1 = 0$, which has a repeated root $x=1$. Substituting $x=1$ into $y^2=4x$ gives $y=\pm 2$. The potential points are $(1, 2)$ and $(1, -2)$. Differentiating $y^2=4x$ gives $\frac{dy}{dx}=\frac{2}{y}$, and differentiating $x^2+y^2-6x+1=0$ gives $\frac{dy}{dx}=\frac{3-x}{y}$. At $(1, 2)$, both derivatives are $1$. At $(1, -2)$, both derivatives are $-1$. Since the derivatives match at both points, the curves touch each other at exactly two points.