1. Consider graphs of two polynomial equations f(x,y) = 0 and g((x,y) = 0, where f and g are both of degree two in two variables. Give specific examples of f and g such that the graphs (zero sets) intersect at exactly 4 point. Sketch the graphs.
2. Consider six different points on an ellipse (like an elliptical lake and 6 points as benches). Join the consecutive pairs of points with lines (i.e. you get a hexagon inside the ellipse). Try to make the sides of the hexagon not parallel, so the lines intersect somewhere. Do you see anything interesting about the points of the intersection?