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Monday 20th of May
Calculations with Negative Numbers
Solving Linear Equations
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Algebra Expressions
Evaluating Expressions and Solving Equations
Fraction rules
Factoring Quadratic Trinomials
Multiplying and Dividing Fractions
Dividing Decimals by Whole Numbers
Adding and Subtracting Radicals
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Slopes of Perpendicular Lines
Linear Equations
Roots - Radicals 1
Graph of a Line
Sum of the Roots of a Quadratic
Writing Linear Equations Using Slope and Point
Factoring Trinomials with Leading Coefficient 1
Writing Linear Equations Using Slope and Point
Simplifying Expressions with Negative Exponents
Solving Equations 3
Solving Quadratic Equations
Parent and Family Graphs
Collecting Like Terms
nth Roots
Power of a Quotient Property of Exponents
Adding and Subtracting Fractions
Solving Linear Systems of Equations by Elimination
The Quadratic Formula
Fractions and Mixed Numbers
Solving Rational Equations
Multiplying Special Binomials
Rounding Numbers
Factoring by Grouping
Polar Form of a Complex Number
Solving Quadratic Equations
Simplifying Complex Fractions
Common Logs
Operations on Signed Numbers
Multiplying Fractions in General
Dividing Polynomials
Higher Degrees and Variable Exponents
Solving Quadratic Inequalities with a Sign Graph
Writing a Rational Expression in Lowest Terms
Solving Quadratic Inequalities with a Sign Graph
Solving Linear Equations
The Square of a Binomial
Properties of Negative Exponents
Inverse Functions
Rotating an Ellipse
Multiplying Numbers
Linear Equations
Solving Equations with One Log Term
Combining Operations
The Ellipse
Straight Lines
Graphing Inequalities in Two Variables
Solving Trigonometric Equations
Adding and Subtracting Fractions
Simple Trinomials as Products of Binomials
Ratios and Proportions
Solving Equations
Multiplying and Dividing Fractions 2
Rational Numbers
Difference of Two Squares
Factoring Polynomials by Grouping
Solving Equations That Contain Rational Expressions
Solving Quadratic Equations
Dividing and Subtracting Rational Expressions
Square Roots and Real Numbers
Order of Operations
Solving Nonlinear Equations by Substitution
The Distance and Midpoint Formulas
Linear Equations
Graphing Using x- and y- Intercepts
Properties of Exponents
Solving Quadratic Equations
Solving One-Step Equations Using Algebra
Relatively Prime Numbers
Solving a Quadratic Inequality with Two Solutions
Operations on Radicals
Factoring a Difference of Two Squares
Straight Lines
Solving Quadratic Equations by Factoring
Graphing Logarithmic Functions
Simplifying Expressions Involving Variables
Adding Integers
Factoring Completely General Quadratic Trinomials
Using Patterns to Multiply Two Binomials
Adding and Subtracting Rational Expressions With Unlike Denominators
Rational Exponents
Horizontal and Vertical Lines
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The Ellipse

An ellipse is the set of points P in the plane such that the sum of the distances from P to two fixed points F1 and F 2 is a constant.
 F1P + F 2P = k the points F1 & F 2 the foci.
Using the idea of the two stacked cones, the ellipse is made when an angled vertical cut is made

The important characteristics of the ellipse:



Example 1
Given 4x 2 + y 2 = 36 , find the equation of the ellipse and give the coordinates of the foci.


The foci

Given the equation, we see the major axis is along the y axis, and is centred at the origin.
This means the vertices are at (0, -6) & (0,6), with co- vertices at (-3,0) & (3,0).

a  = b 2 + c 2
36 - 9  = c 2
27  = c 2
 = c

Hence the foci are at (0,  )  and  (0, )
Ellipse are often not centered at the origin: Hence the possible ellipses are


Sketch , and label the foci
We can from the equation that the major axis is parallel to the x axis.
The length of the major axis is 2a, which is 10. the minor axis is 2b, which is 8.
The vertices are (h - a , k) & (h + a, k), which are going to give ( -6, 3) & (4, 3).
The co-vertices are (h, k - b) & ( h, k + b), which are going to give (-1, -1) & (-1, 7).
The foci are going to be (h - c, k) & ( h + c, k)
Where c is equal to

a  = b 2 + c 2
 c 2  = 25 - 16
c  = 3

Hence the foci are ( -4, 3 ) & ( 2, 3 )
Write the equation of an ellipse in standard form given:
· The centre is at (2,-1).
· The major axis is 16 units and is parallel to the x axis.
· The minor axis is 4 units.
The centre is at (2,-1), with h = 2 & k = -1.
If the length of the major axis is 16, and the major axis is 2a, then

16 = 2a

a = 8

The length of the minor axis is 2.
Since we know h, k, a ,b we can solve the equation.


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