In this lesson, students find points of intersection of a circle (or parabola) and a line. They use graphical methods to determine the intersection points, and then use algebraic methods to verify that the points lie on the circle and line. In this way, students are finding solutions to a system of equations consisting of a linear and a quadratic equation. Finally, they write their own equations that meet certain constraints.

As students write their own equations, they make use of the structure of equations of parallel lines and perpendicular lines (MP7). They continue making connections between the structure of these equations and the graphical representation of lines and circles in the same coordinate plane, as they consider the conditions for points of intersection. 

Technology isn’t required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.