Overview
This lecture covers the concept of circles as a conic section, explains the standard equation of a circle, and demonstrates how to write the equation for circles with various centers and radii.
Conic Sections Overview
- A conic section is formed by the intersection of a plane and a cone.
- The angle of the intersecting plane creates different conic sections; the focus here is on circles.
Definition and Properties of a Circle
- A circle is the set of all points on a plane equidistant from a fixed point called the center.
- The constant distance from the center to any point on the circle is the radius.
Standard Equation of a Circle
- The standard form of a circle's equation: ((x - h)^2 + (y - k)^2 = r^2)
- ( (h, k) ) is the center
- ( r ) is the radius
Examples of Writing Circle Equations
- Center at origin (0, 0) with radius 5: (x^2 + y^2 = 25)
- Center at (0, 3), radius 6: (x^2 + (y - 3)^2 = 36)
- Center at (2, -5), radius 10: ((x - 2)^2 + (y + 5)^2 = 100)
- Center at (1, 5), radius (\sqrt{17}): ((x - 1)^2 + (y - 5)^2 = 17)
- Center at (0, 0), radius 8: (x^2 + y^2 = 64)
- Center at (0, 0), radius (\sqrt{19}): (x^2 + y^2 = 19)
- Center at (0, 2), radius 7: (x^2 + (y - 2)^2 = 49)
- Center at (-7, 5), radius (2\sqrt{14}): ((x + 7)^2 + (y - 5)^2 = 56)
Key Terms & Definitions
- Conic Section — A curve formed by the intersection of a plane and a cone.
- Circle — Set of all points in a plane at a constant distance from a center.
- Center (h, k) — The fixed point from which every point on the circle is equidistant.
- Radius (r) — The constant distance from the center to any point on the circle.
- Standard Form — The equation ((x - h)^2 + (y - k)^2 = r^2).
Action Items / Next Steps
- Practice writing the equation of a circle given different centers and radii.
- Review the standard form and practice identifying center and radius from given equations.