Algebra Tutorials!
Sunday 16th of June  
Exponential Decay
Negative Exponents
Multiplying and Dividing Fractions 4
Evaluating Expressions Involving Fractions
The Cartesian Coordinate System
Adding and Subtracting Fractions with Like Denominators
Solving Absolute Value Inequalities
Multiplying Special Polynomials
FOIL Method
Solving Systems of Equations by Graphing
Graphing Compound Inequalities
Solving Quadratic Equations by Completing the Square
Addition Property of Equality
Square Roots
Adding and Subtracting Fractions
The Distance Formula
Graphing Logarithmic Functions
Dividing Mixed Numbers
Evaluating Polynomials
Power of a Product Property of Exponents
Terminology of Algebraic Expressions
Adding and Subtracting Rational Expressions with Identical Denominators
Solving Exponential Equations
Factoring The Difference of 2 Squares
Changing Fractions to Decimals
Solving Linear Equations
Using Patterns to Multiply Two Binomials
Completing the Square
Roots of Complex Numbers
Methods for Solving Quadratic Equations
Conics in Standard Form
Solving Quadratic Equations by Using the Quadratic Formula
Simplifying Fractions 2
Exponential Notation
Exponential Growth
The Cartesian Plane
Graphing Linear Functions
The Slope of a Line
Finding Cube Roots of Large Numbers
Rotating Axes
Common Mistakes With Percents
Solving an Equation That Contains a Square Root
Rational Equations
Properties of Common Logs
Composition of Functions
Using Percent Equations
Solving Inequalities
Properties of Exponents
Graphing Quadratic Functions
Factoring a Polynomial by Finding the GCF
The Rectangular Coordinate System
Adding and Subtracting Fractions
Multiplying and Dividing Rational Expressions
Improper Fractions and Mixed Numbers
Properties of Exponents
Complex Solutions of Quadratic Equations
Solving Nonlinear Equations by Factoring
Solving Quadratic Equations by Factoring
Least Common Multiples
Solving Exponential Equations
Solving Linear Equations
Multiplication Property of Equality
Multiplying Mixed Numbers
Multiplying Fractions
Reducing a Rational Expression to Lowest Terms
Literal Numbers
Factoring Trinomials
Logarithmic Functions
Adding Fractions with Unlike Denominators
Simplifying Square Roots
Adding Fractions
Equations Quadratic in Form
Dividing Rational Expressions
Slopes of Parallel Lines
Simplifying Cube Roots That Contain Variables
Functions and Graphs
Complex Numbers
Multiplying and Dividing Fractions 1
Composition of Functions
Intercepts of a Line
Multiplying Two Numbers with the same Tens Digit and whose Ones Digits add up to 10
Factoring Trinomials
Exponents and Polynomials
Decimals and their Equivalent Fractions
Negative Integer Exponents
Adding and Subtracting Mixed Numbers
Solving Quadratic Equations
Theorem of Pythagoras
Equations 1
Subtracting Fractions
Solving Quadratic Equations by Graphing
Evaluating Polynomials
Angles and Degree Measure
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Rotating Axes

You probably know already that equations of conics with axes parallel to one of the coordinate axes can be written in the general form

Ax2 + Cy2 + Dx + Ey + F = 0 Horizontal or vertical axes

Here you will study the equations of conics whose axes are rotated so that they are not parallel to the x-axis or the y-axis. The general equation for such conics contains an xy-term.

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 Equation in xy-plane

To eliminate this xy-term, you can use a procedure called rotation of axes. You want to rotate the x- and y-axes until they are parallel to the axes of the conic. (The rotated axes are denoted as the x'-axis and the y'-axis, as shown in the figure below.)

After the rotation has been accomplished, the equation of the conic in the new x'y'-plane will have the form

A'(x')2 + B'x'y' + C'(y')2 + D'x' + E'y' + F' = 0 Equation in x'y'-plane

Because this equation has no x'y'-term, you can obtain a standard form by completing the square.

The following theorem identifies how much to rotate the axes to eliminate an xy-term and also the equations for determining the new coefficients A', C', D', E', and F'.


Rotation of Axes

The general equation of the conic

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

where B ≠ 0, can be rewritten as

A'(x')2 + B'x'y' + C'(y')2 + D'x' + E'y' + F' = 0

by rotating the coordinate axes through an angle θ, where

The coefficients of the new equation are obtained by making the substitutions

x = x' cos θ - y' sin θ

y = x' sin θ + y' cos θ.


To discover how the coordinates in the xy-system are related to the coordinates in the x'y'-system, choose a point P = (x, y) in the original system and attempt to find its coordinates (x', y') in the rotated system. In either system, the distance r between the point P and the origin is the same, and thus the equations for x, y, x', and y'  are those given in the figure below.

Using the formulas for the sine and cosine of the difference of two angles, you obtain

x' = r cos (α - θ) = r (cos α cos θ + sin α sin θ)

= r cos α cos θ + r sin α sin θ = x cos θ + y sin θ

y' = r sin (α - θ) = r (sin α cos θ - cos α sin θ)

= r sin α cos θ - r cos α sin θ = y cos θ - x sin θ

Solving this system for x and y yields

x = x' cos θ - y' sin θ and y = x' sin θ + y' cos θ.

Finally, by substituting these values for x and y into the original equation and collecting terms, you obtain the following.

Now, in order to eliminate the x'y'-term, you must select θ such that B' = 0, as follows.

If B = 0, no rotation is necessary, because the xy-term is not present in the original equation. If B ≠ 0, the only way to make B' = 0 is to let

Thus, you have established the desired results.

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