Conic Sections - Circle Quiz

Study Notes

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.

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.

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

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.

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).