Compare numbers

Learning Objectives Compare the lengths of the major and minor axes of two or more ellipses from their equations. Calculate and compare the eccentricities of different ellipses to determine their relative roundness or elongation. Compare the focal distances of ellipses to understand the placement of their foci. Determine which of two ellipses is larger by comparing their areas. Analyze and compare the standard form equations of ellipses to identify differences in orientation and center. Use the relationship a² = b² + c² to compare the key parameters of an ellipse. Which planet has a more 'squashed' orbit, Earth or Mars? 🪐 We can compare numbers derived from their elliptical paths to find the answer! In this tutorial, we will move beyond simply graphing ellipses....

TermDefinitionExample Major AxisThe longest diameter of an ellipse, passing through its center and both foci. Its length is 2a.For the ellipse (x²/25) + (y²/9) = 1, a² = 25, so a = 5. The major axis has a length of 2a = 10. Minor AxisThe shortest diameter of an ellipse, passing through the center and perpendicular to the major axis. Its length is 2b.For the ellipse (x²/25) + (y²/9) = 1, b² = 9, so b = 3. The minor axis has a length of 2b = 6. Focal Distance (c)The distance from the center of the ellipse to one of its two foci. The total distance between the two foci is 2c.For an ellipse where a=5 and b=3, we find c using c² = a² - b² = 25 - 9 = 16. So, c = 4. Eccentricity (e)A number between 0 and 1 that measures how elongated an ellipse is. An eccentricity of 0 is a perfect circle, while...

Standard Equation of an Ellipse \frac{(x-h)^2}{N_1} + \frac{(y-k)^2}{N_2} = 1 The general form for an ellipse centered at (h, k). The larger denominator is always a², and the smaller is b². If a² is under the x-term, the major axis is horizontal. If a² is under the y-term, the major axis is vertical. Core Parameter Relationship a^2 = b^2 + c^2 This fundamental formula connects the semi-major axis (a), semi-minor axis (b), and the focal distance from the center (c). It is essential for finding one parameter when the other two are known. Eccentricity Formula e = \frac{c}{a} This formula calculates the eccentricity (e). Since c < a for an ellipse, the value of e is always between 0 and 1. A smaller 'e' means the ellipse is more circular.