Conic Sections Overview

Overview

This lecture covers how to identify, graph, and write standard equations for conic sections (circles, ellipses, hyperbolas, and parabolas), including determining their key features from equations.

Circles

• Standard form: (x – h)² + (y – k)² = r², where (h, k) is the center and r is the radius.
• If no (h, k), the center is at the origin.
• To graph: plot the center and mark points r units left, right, up, and down; connect these points.
• If given x² + y² = 9, center is (0,0), radius 3.

Ellipses

• Standard form: (x – h)²/a² + (y – k)²/b² = 1, a > b.
• Center is (h, k); a is under the variable with the major axis.
• From the center, move a units along the major axis and b units along the minor axis.
• Major axis length = 2a; minor axis = 2b.
• Vertices are at endpoints of the major axis; co-vertices at minor axis endpoints.
• Foci (plural of focus): distance from center is c, where c² = a² – b²; along the major axis.

Hyperbolas

• Standard form: (x – h)²/a² – (y – k)²/b² = 1 or (y – k)²/a² – (x – h)²/b² = 1.
• Center at (h, k); transverse axis direction matches the first variable (x: left-right, y: up-down).
• Vertices are a units from the center along the transverse axis.
• Foci: distance from center is c, where c² = a² + b²; located along the transverse axis.
• Asymptote equations: y – k = ±(b/a)(x – h) or y – k = ±(a/b)(x – h), matching the orientation.

Parabolas

• Standard forms: (y – k)² = 4p(x – h) (opens right/left), (x – h)² = 4p(y – k) (opens up/down).
• Vertex at (h, k); p determines distance to focus and directrix.
• Focus: p units from vertex in opening direction.
• Directrix: p units from vertex opposite to opening direction.
• If p > 0, opens up/right; if p < 0, opens down/left.

Identifying Conic Sections

• Circle: x² and y², equal coefficients, same sign.
• Ellipse: x² and y², unequal positive coefficients.
• Hyperbola: x² and y², opposite signs.
• Parabola: only one squared variable.

Putting Equations in Standard Form

• Group x and y terms; factor if necessary.
• Complete the square for x and y terms.
• Adjust the equation to match the standard form.

Key Terms & Definitions

• Center — Point (h, k) about which a conic is symmetric.
• Vertex/Vertices — Endpoints of the major axis (ellipse, hyperbola, parabola).
• Focus/Foci — Special internal points in ellipses, hyperbolas, and parabolas determining shape.
• Directrix — Fixed line used in the definition of a parabola.
• Major axis — The longest axis of an ellipse.
• Minor axis — The shortest axis of an ellipse.
• Transverse axis — Axis connecting vertices in a hyperbola.
• Asymptote — Line that the hyperbola approaches but never touches.
• Completing the Square — Technique for rewriting quadratics to reveal conic properties.

Action Items / Next Steps

• Practice rewriting conic equations into standard form and graphing them.
• Review formulas for foci, axes, and asymptotes for all conic types.
• Complete assigned homework on graphing and identifying conics.