A circle in coordinate geometry connects a familiar shape to algebra, graphing, and measurement. On a coordinate plane, every point on a circle is the same distance from a fixed center. This distance idea becomes an equation that can be graphed, analyzed, and combined with lines or other curves.
Circles matter because they appear in navigation, design, physics, engineering, and many geometry problems.
Key Facts
- Standard form of a circle: (x - h)^2 + (y - k)^2 = r^2
- Center and radius from standard form: center = (h, k), radius = r
- Circle centered at the origin: x^2 + y^2 = r^2
- General form: x^2 + y^2 + Dx + Ey + F = 0
- Distance from center to point on circle: r = sqrt((x - h)^2 + (y - k)^2)
- A tangent line touches a circle at one point and is perpendicular to the radius at that point.
Vocabulary
- Circle
- A circle is the set of all points in a plane that are the same distance from one fixed point.
- Center
- The center is the fixed point inside a circle from which all points on the circle are equally distant.
- Radius
- The radius is the distance from the center of a circle to any point on the circle.
- Tangent line
- A tangent line is a line that touches a circle at exactly one point.
- Secant line
- A secant line is a line that intersects a circle at two points.
Common Mistakes to Avoid
- Forgetting to square the radius is wrong because the standard equation uses r^2, not r, on the right side.
- Reading the center signs incorrectly is wrong because (x - h)^2 + (y - k)^2 means the center is (h, k), so (x + 3)^2 has h = -3.
- Assuming every line that touches the drawing is tangent is wrong because a tangent must intersect the circle at exactly one point algebraically.
- Completing the square without balancing the equation is wrong because adding a value to one side must be matched by adding the same value to the other side.
Practice Questions
- 1 Write the standard equation of the circle with center (3, -2) and radius 5.
- 2 Find the center and radius of the circle x^2 + y^2 - 6x + 8y - 11 = 0.
- 3 A line intersects a circle at exactly one point. Explain why the radius drawn to that point must be perpendicular to the line.