Mathematics
Grade 11
15 min
Find properties of ellipses from equations in general form
Find properties of ellipses from equations in general form
What you'll learn
- Identify whether a whole number (up to 20) is even or odd by dividing it into two equal groups with no remainders, with 80% accuracy.
- Explain why even numbers always end in 0, 2, 4, 6, or 8, and odd numbers always end in 1, 3, 5, 7, or 9, using at least two examples.
- Solve addition problems with two addends (up to 10 each) and determine if the sum is even or odd, with 75% accuracy.
- Apply the rule that "even + even = even" and "odd + odd = even" and "even + odd = odd" to predict whether the sum of two numbers (up to 10 each) will be even or odd, providing a correct justification in 4 out of 5 attempts.
Tutorial Preview
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Introduction & Learning Objectives
Learning Objectives
Identify an equation in general form as an ellipse.
Convert the general form of an ellipse's equation to its standard form by completing the square.
Determine the coordinates of the center (h, k) and the orientation of the major axis.
Calculate the lengths of the major and minor axes.
Find the coordinates of the vertices and co-vertices.
Calculate the focal distance 'c' and determine the coordinates of the foci.
How can a messy equation like 4x² + 9y² - 16x + 18y - 11 = 0 describe the perfect, graceful orbit of a planet? 🪐 Let's find out!
This tutorial will guide you through the process of taking a complicated-looking general form equation and transforming it into the much friendlier standard form of an ellipse. By doing this, you wi...
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Key Concepts & Vocabulary
TermDefinitionExample
General Form of an EllipseAn equation of the form Ax² + Cy² + Dx + Ey + F = 0, where A and C are positive and A ≠ C. This form describes a conic section, but it doesn't immediately reveal the ellipse's properties.9x² + 4y² + 36x - 24y + 36 = 0
Standard Form of an EllipseThe form of an ellipse's equation that clearly shows its properties. It is written as \frac{(x-h)²}{a²} + \frac{(y-k)²}{b²} = 1 or \frac{(x-h)²}{b²} + \frac{(y-k)²}{a²} = 1, where a > b.\frac{(x+2)²}{4} + \frac{(y-3)²}{9} = 1
Completing the SquareAn algebraic technique used to convert a quadratic expression from a general form (ax² + bx) to a perfect square trinomial ((x+d)²). This is the key method for converting the general form of an ellipse to its standard form.To complete the sq...
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Core Formulas
Conversion by Completing the Square
1. Group x-terms and y-terms. Move constant F to the right side.
2. Factor out coefficients A and C from the x and y groups respectively.
3. Complete the square for x and y inside the parentheses.
4. Add the corresponding values to the right side.
5. Divide the entire equation by the new constant on the right to make it equal to 1.
This is the step-by-step process to transform the general form Ax² + Cy² + Dx + Ey + F = 0 into the standard form \frac{(x-h)²}{a²} + \frac{(y-k)²}{b²} = 1.
Focal Length Formula
c² = a² - b²
Use this formula to find the distance 'c' from the center to each focus. Remember that a² is always the larger denominator in the standard form equation.
Coordinates of Key Points
Center: (h, k)
Horizonta...
4 more steps in this tutorial
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Challenging
The equation of an ellipse is 4x² + 9y² - 8x + 36y + k = 0. If the center of the ellipse is at (1, -2), what must be the value of k?
A.4
B.36
C.-4
D.40
Challenging
Consider Ellipse A: x² + 4y² - 6x + 16y + 21 = 0 and Ellipse B: 4x² + y² + 8x - 6y + 9 = 0. Which statement is true?
A.Ellipse A has a longer major axis than Ellipse B.
B.Ellipse B has a longer major axis than Ellipse A.
C.Both ellipses have the same major axis length.
D.Both ellipses have the same center.
Challenging
Which of the following is NOT a property of the ellipse defined by 16x² + 9y² + 64x - 18y - 71 = 0?
A.The center is at (-2, 1).
B.The length of the major axis is 8.
C.The foci are at (-2, 1 ± √7).
D.The co-vertices are at (1, 1) and (-5, 1).
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Frequently asked questions
What grade level is "Find properties of ellipses from equations in general form"?
Find properties of ellipses from equations in general form is a Grade 11 Mathematics lesson on ExcelOS.
What will I learn in Find properties of ellipses from equations in general form?
You'll be able to: Identify whether a whole number (up to 20) is even or odd by dividing it into two equal groups with no remainders, with 80% accuracy; Explain why even numbers always end in 0, 2, 4, 6, or 8, and odd numbers always end in 1, 3….
Is "Find properties of ellipses from equations in general form" free to practice?
Yes. You can read the tutorial preview for free, and signing up for a free ExcelOS account unlocks the full tutorial and all practice questions with instant feedback.
How many practice questions are included with Find properties of ellipses from equations in general form?
This lesson includes 37 practice questions across multiple difficulty levels, each with instant feedback and explanations.