SAT MATH Circle Geometry

Question

How many points may be contained in the intersection of 2 distinct circles?

The answer is 0, 1 and 2 points. I don't get why it could be 0 points when two circles are intersecting.

@DeepSea: Not necessarily. ... in mathematics the intersection of two things can definitely sometimes be the empty set, so no points in common. — Ted Shifrin, May 13 '17 at 04:30
I think the question was too ambiguous, but as Jose answered, concentric circles could be counted for $0$ points. — infinitylord, May 13 '17 at 04:38
The question is not ambiguous. Take note, it asks: "How many points may be contained in the intersection of $2$ distinct circles?" and NOT "How many points may be contained in the intersection of $2$ distinct intersecting circles?" — Jose Arnaldo Bebita Dris, May 13 '17 at 04:46
The problem is asking how many different amount of points can fit. If you draw 2 circle intersecting, it has an intersecting point, but it doesn't need to be counted. $0$ points could "fit" in this region. — Eric Lee, May 13 '17 at 05:15

score 4 · Answer 1 · answered May 13 '17 at 04:33

4

No points of intersection

One point of intersection

Two points of intersection

answered May 13 '17 at 04:33

but in case 1 and 2 the the circles are not intersect with each other – k.Vijay May 13 '17 at 04:40
@k.Vijay, please see Ted Shifrin's and Lê Đức Minh's comments above. – Jose Arnaldo Bebita Dris May 13 '17 at 04:42
It is possible to give examples from analytic geometry where these cases become apparent and clear. I will edit those examples into this answer when time permits. – Jose Arnaldo Bebita Dris May 13 '17 at 04:44

score 1 · Accepted Answer · answered May 13 '17 at 04:57

The problem clearly asks:

"How many points may be contained in the intersection of 2 distinct circles?"

And thus, the assumption that the circles are actually intersecting in all three cases is wrong.

There can be $0$ points in the intersection of two distinct circles, if they are not intersecting. This implies that there is a distance between any point on circle $A$ and any point on circle $B$.
There is $1$ point of intersection, when there is only one point $P$ that is included in both circles.
There are $2$ points of intersection, when only two points are common to both circles.

See the image below for clarifications:

2 Answers2