Conic Sections - Circle Quiz

Conic Sections - Circle

• Conic sections are formed by intersecting a cone and a plane.
• A circle is a conic section formed when the plane intersects the cone perpendicular to its axis of symmetry.
• A circle is a set of points equidistant from a central point.

Basic Concepts of a Circle

• Center: The central point from which all points on the circle are equidistant.
• Circumference: The curve that forms the boundary of the circle.
• Diameter: A line segment passing through the center and connecting two points on the circumference.
• Radius: A line segment drawn from the center to a point on the circumference. The radius is half the length of the diameter.
• Chord: A line segment connecting two points on the circumference.
• Arc: A continuous portion of the circle's circumference between two points.

Standard Equation of a Circle

• The standard equation of a circle with center (h, k) and radius r is:
  • (x - h)² + (y - k)² = r²

Graphing a Circle

• Step 1: Determine the values of h, k, and r from the equation.
• Step 2: Locate the center (h, k) on the coordinate plane.
• Step 3: Determine the endpoints of the horizontal diameter (h ± r, k) and the vertical diameter (h, k ± r).
• Step 4: Draw the circle passing through these four points and label the center, radius, and diameter endpoints.

Expressing the Polynomial Equation of a Circle in Standard Form

• Step 1: Transpose the constant term to the right side of the equation.
• Step 2: Group the x terms and y terms together.
• Step 3: Complete the square for both the x terms and y terms.
• Step 4: Express the completed squares in terms of (x + a/2)² and (y + b/2)², and simplify the constant term on the right side.
• For the general equation:
  • x² + y² + ax + by + c = 0
• The standard form is:
  • (x + a/2)² + (y + b/2)² = (a²/4) + (b²/4) - c
• The center is at (-a/2, -b/2) and the radius is √((a²/4) + (b²/4) - c).