What is the maximum number of points of intersection of two different lines and three different circles in the same plane?
the three circles can cut each other in 6 different points (think of a typical Venn diagram)
A line can cut a circle in 2 point
so each line can cut all the circles in 6 points
so total = 6 + 6 + 6 = 18
BUT, let's not forget that the two lines can also intersect
so the total is 19
To find the maximum number of points of intersection between two lines and three circles in the same plane, we can calculate the number of intersections for each pair of objects and then add them up.
1. Lines:
- Two lines can intersect at a maximum of one point.
2. Circles:
- Two circles can intersect at a maximum of two points, as there can be two intersection points along their common chord.
- Three circles can intersect pairwise, resulting in a total of three pairwise intersections.
Now, let's calculate the total number of intersections:
- For lines: Since we have two lines, the maximum number of intersections is 1.
- For circles: Since we have three circles, the maximum number of pairwise intersections is 3.
Adding the intersections of lines and circles, we get:
1 (intersections between lines) + 3 (pairwise intersections between circles) = 4.
Therefore, the maximum number of points of intersection between two different lines and three different circles in the same plane is 4.