Analytic Geometry: Conic Sections

Introduction to Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. The four main types are parabolas, circles, ellipses, and hyperbolas. Each has a standard equation and unique geometric properties. Recognizing and converting equations to standard form is essential for graphing and analysis.

General Equation of a Conic Section

• The general second-degree equation for a conic section is:

• The type of conic depends on the coefficients A, B, and C:

• Circle: ,

• Ellipse: , and have the same sign,

• Parabola: Either or , but not both

• Hyperbola: and have opposite signs,

Standard Forms of Conic Sections

• Circle:

• Center:

• Radius:

• Ellipse:

• Center:

• Major axis: (if ), Minor axis:

• Hyperbola:

(opens left/right) (opens up/down)

• Center:

• Vertices: units from center along transverse axis

• Parabola:

(opens up/down) (opens left/right)

• Vertex:

• Axis of symmetry: parallel to x- or y-axis

Identifying and Converting Equations

• To identify the conic, compare the coefficients of and .

• To convert to standard form, complete the square for and terms as needed.

Example 1:

Given

• Group and terms:

• Complete the square for each variable:

• This is a circle with center and radius .

Example 2:

Given

• Rewrite:

• Complete the square for :

• This is a parabola opening to the right with vertex .

Key Properties and Classification Table

Graph Identification and Matching

• To match equations to graphs, identify the conic type and its orientation (e.g., opens up/down or left/right).

• For ellipses and hyperbolas, compare denominators to determine axis lengths and orientation.

• For parabolas, note the direction it opens and the vertex location.

Example: Matching Equation to Graph

• Given , this is an ellipse centered at the origin, major axis along the x-axis.

• Given , this is a hyperbola centered at the origin, opening up and down.

Summary of Steps for Analyzing Conic Sections

1. Identify the conic: Compare coefficients of and .

2. Rewrite in standard form: Complete the square as needed.

3. Find key features: Center, vertex, axes, direction of opening.

4. Match to graph: Use standard form and features to select the correct graph.

Practice Problems (from file)

• Identify each conic section and write in standard form. For parabolas, state the vertex and direction. For ellipses and hyperbolas, state the center, axes, and orientation.

• Match equations to graphs and descriptions as shown in the provided questions.

Additional info: The above notes synthesize the main concepts and procedures needed to answer the exam-style questions shown in the file, focusing on analytic geometry and conic sections as covered in College Algebra.