Mathematics
Grade 11
15 min
Order numbers
Order numbers
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Identify the key numerical parameters (a, b, c) of an ellipse from its standard equation.
Calculate the focal distance (c) and eccentricity (e) of an ellipse.
Order the values of a, b, and c for any given ellipse and explain their fundamental relationship (a > b and a > c).
Correctly place the eccentricity (e) in order relative to 0 and 1 (0 < e < 1) and explain what its value signifies about the ellipse's shape.
Compare and order multiple ellipses based on their eccentricity.
Order the lengths of the major axis (2a), minor axis (2b), and the interfocal distance (2c).
Ever wondered why planetary orbits are described as 'elliptical' and not just 'oval'? 🪐 It all comes down to a precise ordering of a few key numbers tha...
2
Key Concepts & Vocabulary
TermDefinitionExample
Semi-major Axis (a)The distance from the center of the ellipse to a vertex (one of the two farthest points on the ellipse). It is always the longest radius.In the ellipse given by the equation x²/25 + y²/16 = 1, a² = 25, so the semi-major axis a = 5.
Semi-minor Axis (b)The distance from the center of the ellipse to a co-vertex (one of the two closest points on the ellipse). It is always the shortest radius.In the ellipse given by the equation x²/25 + y²/16 = 1, b² = 16, so the semi-minor axis b = 4.
Focal Distance (c)The distance from the center of the ellipse to a focus (one of two special fixed points inside the ellipse).For the ellipse x²/25 + y²/16 = 1, we find c using c² = a² - b² = 25 - 16 = 9. So, the focal distance c = 3.
The Fundamental Order of Ellipse Leng...
3
Core Formulas
Standard Equation of an Ellipse
\frac{(x-h)^2}{n_1} + \frac{(y-k)^2}{n_2} = 1
This is the general form for an ellipse centered at (h, k). The key rule for ordering is that a² is ALWAYS the larger denominator, and b² is the smaller one. If the larger denominator is under the x-term, the major axis is horizontal. If it's under the y-term, the major axis is vertical.
Focal Distance Formula
c^2 = a^2 - b^2
This formula relates the three primary lengths of an ellipse. It is used to calculate the focal distance 'c' once 'a' and 'b' are known from the standard equation. This formula guarantees that a > c.
Eccentricity Formula
e = \frac{c}{a}
This formula calculates the eccentricity 'e'. Since we know a > c for any ellipse,...
4 more steps in this tutorial
Sign up free to access the complete tutorial with worked examples and practice.
Sign Up Free to ContinueSample Practice Questions
Challenging
Ellipse A has a semi-major axis a_A = 15 and semi-minor axis b_A = 9. Ellipse B has a focal distance c_B = 16 and an eccentricity e_B = 0.8. Which statement correctly orders their eccentricities?
A.e_B > e_A
B.e_A > e_B
C.e_A = e_B
D.The order cannot be determined.
Challenging
An ellipse is centered at the origin with its major axis on the y-axis. Its semi-minor axis is b=6 and its eccentricity is e=4/5. What is the correct ordering of a, b, and c?
A.c < b < a
B.b < c < a
C.b < a < c
D.a < c < b
Challenging
Given the general equation of an ellipse centered at (h, k), (x-h)²/n + (y-k)²/m = 1, where n and m are positive constants and n > m. Which expression correctly represents the eccentricity 'e' in terms of n and m?
A.√(n - m) / m
B.(n - m) / n
C.√(n - m) / √n
D.√n / √(n - m)
Want to practice and check your answers?
Sign up to access all questions with instant feedback, explanations, and progress tracking.
Start Practicing Free