View source for What is the greatest number of points of intersection that can occur when $2$ different circles and $2$ different straight lines are drawn on the same piece of paper?

What is the greatest number of points of intersection that can occur when <math>2</math> different circles and <math>2</math> different straight lines are drawn on the same piece of paper?






Solution 1:
Make a diagram. Two geometric figures intersect if they have one or more points in common. Draw two circles which intersect in <math>2</math> points. Draw a line which intersects the two circles in <math>4</math> points. Draw another line which intersects the two circles in <math>4</math> points and also intersects the first line. There are <math>\boxed{11}</math> points of intersection.[asy]

draw(Circle((-0.7,0),1));
draw(Circle((0.7,0),1));

dot((0,0));

dot((0,0.7));
dot((0,-0.7));

draw((0,0)--(-2,0.6),Arrow);
draw((0,0)--(-2,-0.6),Arrow);
draw((0,0)--(2,0.6),Arrow);
draw((0,0)--(2,-0.6),Arrow);

dot((-1.58,0.47));
dot((-1.58,-0.47));
dot((1.58,0.47));
dot((1.58,-0.47));

dot((-0.29,0.08));
dot((-0.29,-0.08));
dot((0.29,0.08));
dot((0.29,-0.08));

[/asy]
Solution 2:
Make a table of the maximum number of points of intersection.<cmath>
\begin{array}{|c|c|}
\hline
\text{Geometric Figures} & \text{Number of Common Points} \\
\hline
\text{2 Circles} & 2 \\
\hline
\text{Line (1) and 2 circles} & 4 \\
\hline
\text{Line (2) and 2 circles} & 4 \\
\hline
\text{2 lines} & 1 \\
\hline
& \text{Total = \boxed{11}} \\
\hline
\end{array}
</cmath>