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?

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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?




Solution 1: Make a diagram. Two geometric figures intersect if they have one or more points in common. Draw two circles which intersect in $2$ points. Draw a line which intersects the two circles in $4$ points. Draw another line which intersects the two circles in $4$ points and also intersects the first line. There are $\boxed{11}$ 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.\[\begin{array}{|c|c|} \hline \text{Geometric Figures} & \text{Number of Common Points} \\ \hline \text{2 Circles} & 2 \\ \hline \text{Line (1) and 2 circles} & 4 \\ \hline \text{Line (2) and 2 circles} & 4 \\ \hline \text{2 lines} & 1 \\ \hline & \text{Total = \boxed{11}} \\ \hline \end{array}\]