Conic Sections

Ellipses

An ellipse is the set of all points in a plane such that the sum of the distances from two fixed points (foci) is constant.

• Standard Equation (centered at (h, k)):

• Major axis: The longer axis; length =

• Minor axis: The shorter axis; length =

• Foci: Located at or where

• Example: is an ellipse centered at (0,0), major axis along y-axis.

Circles

A circle is a set of all points in a plane that are a fixed distance (radius) from a fixed point (center).

• Standard Equation (centered at (h, k)):

• Example: is a circle centered at (2, -1) with radius 3.

Parabolas

A parabola is the set of all points equidistant from a fixed point (focus) and a fixed line (directrix).

• Standard Equation (vertical): or

• Vertex:

• Focus:

• Directrix:

• Axis of symmetry:

• Example: can be rewritten and graphed as a parabola.

Hyperbolas

A hyperbola is the set of all points in a plane such that the absolute value of the difference of the distances from two fixed points (foci) is constant.

• Standard Equation (centered at (h, k)): (opens left/right) (opens up/down)

• Vertices: or

• Foci:

• Asymptotes:

• Example: is a hyperbola opening left/right.

Sequences and Series

Arithmetic Sequences

An arithmetic sequence is a sequence where each term after the first is found by adding a constant difference to the previous term.

• General term:

• Sum of first n terms:

• Example: For , ,

Geometric Sequences

A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a constant ratio.

• General term:

• Sum of first n terms: (for )

• Sum of infinite series: (for )

• Example: , ,

Special Sequences and Sums

• Finding nth term: Use the general formula for arithmetic or geometric sequences.

• Partial sums: Calculate using the sum formulas above.

• Example: (geometric series with , )

Binomial Theorem and Pascal's Triangle

Binomial Theorem

The Binomial Theorem provides a formula for expanding powers of binomials.

• Formula:

• Binomial Coefficient:

• Example: Expand using coefficients from Pascal's Triangle:

Pascal's Triangle

Pascal's Triangle is a triangular array of numbers used to find binomial coefficients.

• Each number is the sum of the two numbers directly above it.

• Used to expand binomials and find coefficients quickly.

Finding Specific Terms in Binomial Expansions

• To find the term containing in , use:

• Example: The term containing in is:

Summary Table: Conic Sections

Additional info: These notes are based on exam review questions and solutions, covering conic sections, sequences, series, and binomial expansions, all of which are core topics in College Algebra.