Line Intersects Circle at Most Twice

The cases for 0, 1, and 2 intersections, respectively, no intersection, tangent, and secant, are all fairly well defined. So the question is: Can a line intersect a circle at more than 2 points?

We prove by contradiction. Suppose we have a circle with three or more collinear points. We then rotate the circle such that those points lie parallel to the x-axis, thus determining the y-value of our intersecting (horizontal) line which we can write as: $$y=n$$

Now we look at the equation of a circle: $$(x-h)^{2}+(y-k)^{2}=r^{2}$$ From our analysis, we already know the values of h, y, k,and r, or rather they are constants as opposed to variables. Taking all non-x to be constant, this leaves us with the general equation: $$ax^{2}+bx+c=0$$ By the fundamental theorem of algebra, there are exactly 2 roots and by extension at most 2 distinct roots by virtue of the polynomial having an order of 2. However, if our line intersected the circle at more than 2 points then the solution set must contain at least 3 distinct roots (otherwise the overlap point would be redundant) which would violate the fundamental theorem of algebra. Thus, we have arrived at a contradiction.

Therefore, a line cannot intersect a circle at more than two points.

Typically, given a line in a plane and a circle, the line will intersect the circle at $0$ or $2$ points.

Let's take the case when a line hits a circle at two points. Say the line hits the circle at $A,B$. Then $AB$ is called a chord of the circle. The Diameter of a circle, which you should be familiar with, is the longest chord (it is also the chord going through the center of the circle).

Note that chords are line segments. Secants are lines. So line $AB$ is a secant of the circle.

Radii is the plural form of radius. For your info, this is a radius:

A circle can have many radii.

A tangent touches a circle at one point.

You can see from the diagram that the tangent touches the circle at point B.

A line can intersect a circle in 0, 1, or 2 points.

A line is either a secant, a tangent, or has no relationship with a circle. If it has no relationship it obviously has no intersections, and if it is tangent it has 1 intersection. If it is a secant, it has 2 intersections. Thus the maximum is 2.

Chords are segments with endpoints in the circle as shown below in red:

A chord can also be a diameter.

Secants are lines that pass through the circle exactly twice.

A chord is a line totally inside a circle, with no points at all outside the circle's circumference. Here is a picture of a chord $XY$:

A secant is a line running right through a circle, going through two points on its circumference. Line $AY$ is a secant. Last of all, tangents are lines crossing $1$ and only $1$ point on a circle's circumference. Line $XD$ is a tangent.