Overview
This lecture covers the basics of circles within the conic sections, focusing on definitions, standard equations, and finding the equation of a circle given its center and radius.
Conic Sections Overview
- A conic section is formed by the intersection of a plane and a cone.
- Four types of conic sections exist, but the focus here is on circles.
Definition of a Circle
- A circle consists of all points in a plane that are equidistant from a fixed point called the center.
- The constant distance from the center to any point on the circle is called the radius.
Standard Form Equation of a Circle
- The standard form is: ((x - h)^2 + (y - k)^2 = r^2), where (h, k) is the center and r is the radius.
- If the center is at the origin (0, 0), the equation simplifies to (x^2 + y^2 = r^2).
Example Problems
- Center at (0,0), radius 5: Equation is (x^2 + y^2 = 25).
- Center at (0,3), radius 6: Equation is (x^2 + (y-3)^2 = 36).
- Center at (2,-5), radius 10: Equation is ((x-2)^2 + (y+5)^2 = 100).
- Center at (1,5), radius (\sqrt{17}): Equation is ((x-1)^2 + (y-5)^2 = 17).
- Center at (0,0), radius 8 ((r^2)=64): Equation is (x^2 + y^2 = 64).
- Center at (0,0), radius (\sqrt{19}): Equation is (x^2 + y^2 = 19).
- Center at (0,2), radius 7 ((r^2)=49): Equation is (x^2 + (y-2)^2 = 49).
- Center at (-7,5), radius (2\sqrt{14}) ((r^2)=56): Equation is ((x+7)^2 + (y-5)^2 = 56).
Key Terms & Definitions
- Circle — set of all points equidistant from a fixed center point.
- Center (h, k) — the fixed point equidistant to every point on the circle.
- Radius (r) — constant distance from the center to any point on the circle.
- Standard Form — ((x-h)^2 + (y-k)^2 = r^2), the general equation of a circle.
Action Items / Next Steps
- Practice writing equations of circles given different centers and radii.
- Review notes on the standard form of the circle equation.