Which number is greatest/least?

Learning Objectives Identify the greatest and least values for the coordinates of vertices and co-vertices of an ellipse. Compare the lengths of the major and minor axes of two or more ellipses to determine which is greatest. Calculate the distance from the center to a focus (c) for multiple ellipses and identify the greatest or least value. Compute the eccentricity of different ellipses and determine which is greatest (most elongated) or least (most circular). Compare the areas of two or more ellipses. Determine the greatest and least distance from a focus to a point on the ellipse. Which planet in our solar system has the most 'squashed' orbit, and which has the most circular? 🪐 We can answer this by finding the greatest and least values associated with elliptic...

TermDefinitionExample Major AxisThe longest diameter of an ellipse, passing through its center and two vertices. Its length is 2a.For the ellipse (x^2/25) + (y^2/9) = 1, a^2 = 25, so a = 5. The major axis has the greatest length, which is 2a = 10. Minor AxisThe shortest diameter of an ellipse, passing through its center and two co-vertices. Its length is 2b.For the ellipse (x^2/25) + (y^2/9) = 1, b^2 = 9, so b = 3. The minor axis has the least length, which is 2b = 6. Foci (plural of Focus)Two fixed points inside an ellipse. The sum of the distances from any point on the ellipse to the two foci is constant and equal to the major axis length (2a).The distance from the center to each focus is 'c'. A greater 'c' value means the foci are farther from the center. VerticesTh...

Standard Ellipse Equations Horizontal: \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \quad Vertical: \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 Used to find the center (h, k) and the values of 'a' and 'b'. Remember, a^2 is always the greater denominator. If a^2 is under the x-term, the ellipse is horizontal; if it's under the y-term, it's vertical. Focal Length Relationship c^2 = a^2 - b^2 This formula calculates 'c', the distance from the center to a focus. A greater 'c' value means the foci are more spread out. This is essential for comparing focal distances. Eccentricity Formula e = \frac{c}{a} Use this to calculate the eccentricity. The value of 'e' for an ellipse is always between 0 and 1 (0 ≤ e <...