A tangent to a circle touches the circle at exactly one point. A tangent line contains at most how many chords of that circle?

Introduction / Context:
This question tests understanding of the concepts of a tangent and a chord in circle geometry. A tangent is a line that touches a circle at exactly one point, while a chord is a line segment whose endpoints both lie on the circle. The problem asks how many chords a tangent line can contain at most. This requires careful distinction between a full line and a line segment that lies inside the circle.

Given Data / Assumptions:

• We have a fixed circle in the plane.
• A tangent to this circle is a straight line that touches the circle at exactly one point.
• A chord is defined as a segment that has both endpoints on the circle.
• We are asked for the maximum possible number of distinct chords that lie entirely on the tangent line.
• We assume standard Euclidean definitions without degeneracy.

Concept / Approach:
To determine how many chords can lie on a tangent, we must recall that for a chord to exist, both its endpoints must be distinct points on the circle, and the interior of the segment lies inside the circle. A tangent line intersects the circle at a single point, not two. Therefore it cannot provide two distinct points on the circle for a chord. At best, it has one common point with the circle, and a chord requires two. Thus the number of chords on a tangent line is zero.

Step-by-Step Solution:
Step 1: Recall the definition of a chord. A chord is any line segment with both endpoints on the circle.Step 2: Recall the definition of a tangent. A tangent to a circle touches the circle at exactly one point.Step 3: For a line to contain a chord of the circle, it must intersect the circle at two distinct points, which will be the endpoints of the chord.Step 4: A tangent line, by definition, intersects the circle at only one point, so there is no second distinct point of intersection.Step 5: Since there is no pair of distinct intersection points, there is no possible chord whose entire segment lies on the tangent line.Step 6: Therefore the maximum number of chords of the circle that the tangent can contain is zero.

Verification / Alternative check:
We can compare this with a secant line, which cuts the circle at exactly two points. A secant line does contain one chord: the segment joining those two intersection points. In contrast, a tangent line is the limiting case where the two intersection points merge into one. As the secant line rotates to become tangent, the chord shrinks to a single point and is no longer considered a chord. This visualization supports the conclusion that a true tangent line does not contain any nondegenerate chord.

Why Other Options Are Wrong:
Option a suggests that there could be exactly one chord on the tangent line, but that would require two distinct intersection points, which contradicts the definition of a tangent. Option b and option c suggest even more chords, which is impossible since even one chord cannot exist on a line that touches at only one point. Thus all these options conflict with the fundamental geometric definitions.

Common Pitfalls:
Some learners misinterpret the single point of contact as a very short chord. However, a chord must have length and two distinct endpoints. A single point has no length and does not qualify. Another pitfall is confusing the behavior of secants with tangents. Remember that tangents are special precisely because they intersect the circle at only one point.

Final Answer:
A tangent to a circle contains at most 0 chords of that circle.