In this activity, students graph a circle and line, and then use the graph to identify points of intersection. They also verify these intersection points using algebraic methods. This is an example of solving a system consisting of a linear equation and a quadratic equation in two variables graphically, although students are not expected to use that language to describe their process in this lesson. 

Monitor for students who use different strategies to verify that the intersection points lie on both the line and the circle. Some approaches students may use include:

• Substitute the points directly into the circle equation, and then verify that the equation is true
• Use the Pythagorean Theorem to determine that the distance from the point to the center of the circle is the same as the radius of the circle.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Invite previously identified students to share their strategies for verifying that the points are on both the line and the circle. If possible, invite one who used the Pythagorean Theorem and one who substituted the point into the circle equation. Ask students, “Why do both of these methods work?” (They are both basically saying the same thing. We need to verify that the point is 5 units away from the center . Any point that satisfies the circle equation meets this description. The Pythagorean Theorem provides the distance directly.)

Students apply their understanding of slopes of parallel and perpendicular lines to problems involving lines and circles. They also consider possible intersections of circles and lines. As students create their own equations that meet particular conditions, they are making use of the structure of the equations of parallel and perpendicular lines and of equations of circles (MP7).

Monitor for students using different strategies to identify the second intersection point of the line and circle. Some approaches students may use include:

• Graphing the line and circle and identifying the point of intersection on the graph.
• Verifying that the line has a -intercept at or rewriting the equation in slope-intercept form to see the intercept.
• Using a slope triangle to identify points on the line until they find one that is also on the circle.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

If student graphs aren’t accurate enough to find the second intersection point in the last problem, suggest that they use slope triangles to find several points on the line until they find one that is also on the circle.

If possible, invite a student who rewrote the third equation in slope-intercept form to share that method. If no student did this, ask the class to do so now. Then ask students:

• “We can rewrite the line as . What does that tell you about where the circle and line intersect?” (The -intercept of the line is . That point is also on the circle, so it’s the second intersection point.)
• “How could you find the exact points where your horizontal line intersects the circle?” (Substitute the particular value of into the circle’s equation and solve for .)
• “What is an equation for a line that intersects the circle in exactly 1 point?” ( )