1956 AMC 12 Problem 21

Question

If each of two intersecting lines intersects a hyperbola and neither line is tangent to the hyperbola, then the possible number of points of intersection with the hyperbola is:

2

2

or

3

2

or

4

3

or

4

2,

3,

or

4

Answer: E

Concepts:hyperbola line-conic intersections counting distinct points

Difficulty rating: 1770

Small hint:

A non-tangent line that intersects a hyperbola can meet it in either one or two points

Big hint:

The two lines may share one hyperbola point, since the lines themselves intersect

Solution:

A line can meet a hyperbola in either one finite point, when its direction is parallel to an asymptote, or two finite points. By choosing the two lines so that their finite intersection sets on the hyperbola are disjoint, they can therefore contribute

1+1=2,

1+2=3,

or

2+2=4

distinct intersections. If their own intersection lies on the hyperbola, one point is shared instead.

Thus,

2,

3,

or

4

are possible, and the correct answer is E.