Graph the line . At what points does this line appear to intersect the circle?
Verify that the 2 figures really intersect at these points. Be prepared to explain your reasoning.
Graph the line . At what points does this line appear to intersect the circle? Verify that the 2 figures really do intersect at these points.
4.3
Activity
Creating Lines
Standards Alignment
Building On
Addressing
A-REI.7
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x² + y² = 3.
Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
Write an equation representing the circle in the graph.
Graph and write equations for each line described:
any line parallel to the -axis that intersects the circle at 2 points
any line perpendicular to the -axis that doesn’t intersect the circle
the line perpendicular to that intersects the circle at
For the last line you graphed, find the second point where the line intersects the circle. Explain or show your reasoning.
Student Lesson Summary
We can graph circles and lines on the same coordinate grid and estimate where they intersect. The image shows the circle and the line . The two figures appear to intersect at the points and . To verify whether these truly are intersection points, we can check if substituting them into each equation produces true statements.
Let’s test . First, substitute it into the equation for the line. When we do so, we get . This is a true statement, so this point is on the line.
Next, substitute it into the equation for the circle. This is the same as checking to see if the distance from the point to the center is , or 13 units. We get . Evaluate the left side to get . This is a true statement, so the point is on the circle. It’s on both the circle and the line, so it must be an intersection point for the two figures.
Glossary
None
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Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x² + y² = 3.
Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x² + y² = 3.