Conic Sections

Circles

A circle is the set of all points in a plane that are equidistant from a fixed point called the center. The distance from the center to any point on the circle is called the radius.

• Standard Equation: The equation of a circle with center and radius is $

• Determining Center and Radius: Given the equation of a circle, rewrite it in standard form to identify the center and radius.

• Graphing: Plot the center and use the radius to draw the circle.

• Application: Circles are used in geometry, physics, and engineering to model round objects and paths.

Parabolas

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

• Standard Equation: For a parabola opening right/left: $

• Vertex, Focus, Directrix: The vertex is , the focus is or depending on orientation, and the directrix is or .

• Graphing: Plot the vertex, focus, and directrix, then sketch the parabola.

• Applications: Parabolas model projectile motion and satellite dishes.

Ellipses

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

• Standard Equation: where is the center, is the semi-major axis, and is the semi-minor axis.

• Vertices and Foci: Vertices are at or ; foci are at where .

• Graphing: Plot the center, vertices, and foci, then sketch the ellipse.

• Applications: Ellipses model planetary orbits and optics.

Hyperbolas

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

• Standard Equation: or

• Vertices, Foci, Asymptotes: Vertices are at or ; foci at where . Asymptotes are lines through the center with slopes .

• Graphing: Plot the center, vertices, foci, and asymptotes, then sketch the hyperbola.

• Applications: Hyperbolas model radio navigation and certain optical systems.

Systems of Equations and Matrices

Solving Systems of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. Solutions can be found using substitution, elimination, or matrix methods.

• Substitution and Elimination: Solve for one variable and substitute into other equations, or add/subtract equations to eliminate variables.

• Classification: Systems can be consistent (at least one solution), inconsistent (no solution), or independent (unique solution).

• Application Problems: Systems model real-world scenarios such as business, chemistry, and physics.

Matrices and Row Operations

A matrix is a rectangular array of numbers. Matrices are used to represent and solve systems of equations.

• Row Echelon Form: Use elementary row operations to transform a matrix to row echelon form (REF) or reduced row echelon form (RREF).

• Inverse of a Matrix: The inverse of a matrix is such that , where is the identity matrix.

• Multiplication: Matrix multiplication is not commutative; may not equal .

• Determinant: The determinant is a scalar value that can be computed for square matrices and is used to determine invertibility.

Solving Linear Systems with Matrices

• Augmented Matrix: Write the system as an augmented matrix and use row operations to solve.

• Inverse Method: If is invertible, solve by .

• Application: Used in engineering, computer science, and economics.

Properties and Operations of Matrices

• Matrix Arithmetic: Add, subtract, and multiply matrices; multiply by scalars.

• Special Matrices: Identity, diagonal, triangular, zero, and inverse matrices.

• Determinant Calculation: Use cofactor expansion or row reduction.

Partial Fractions

Decomposition of Rational Expressions

Partial fraction decomposition is the process of expressing a rational function as a sum of simpler fractions. This is useful for integration and solving equations.

• Linear Factors: Decompose expressions with non-repeated and repeated linear factors in the denominator.

• Quadratic Factors: Decompose expressions with irreducible quadratic factors, repeated or non-repeated.

• General Form: where is factored into linear and quadratic terms.

• Application: Used in calculus for integration and in engineering for signal processing.

Summary Table: Conic Sections