Write equations of ellipses in standard form

Learning Objectives Identify the center, vertices, co-vertices, and foci of an ellipse from given information. Determine if an ellipse has a horizontal or vertical major axis. Calculate the values of a, b, and c for an ellipse. Apply the standard form equations for ellipses centered at (h, k). Write the equation of an ellipse in standard form given its key features (e.g., vertices and foci). Derive the equation of an ellipse from a graphical representation. Ever wondered why a 'whispering gallery' works or how planets orbit the sun in a predictable path? 🪐 The secret lies in the elegant geometry of the ellipse! This tutorial will guide you through the process of translating the geometric properties of an ellipse—like its center, vertices, and foci—into a standard...

TermDefinitionExample EllipseA set of all points (x, y) in a plane, the sum of whose distances from two distinct fixed points, called foci, is constant.If the foci are at (-3, 0) and (3, 0), and the constant sum is 10, a point like (5, 0) is on the ellipse because the distance to (-3, 0) is 8 and the distance to (3, 0) is 2, and 8 + 2 = 10. Center (h, k)The midpoint of the major axis, the minor axis, and the segment connecting the foci.For an ellipse with vertices at (2, 1) and (10, 1), the center is the midpoint, which is (6, 1). Major AxisThe longer axis of the ellipse, which passes through the center and the two vertices. Its length is 2a.If an ellipse has vertices at (0, 5) and (0, -5), its major axis is the vertical segment between them, with a length of 10 units (so a = 5). Minor Ax...

Standard Form of an Ellipse (Horizontal Major Axis) \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 Use this form when the ellipse is wider than it is tall. The larger denominator, a^2, is under the x-term. Here, a > b. Standard Form of an Ellipse (Vertical Major Axis) \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 Use this form when the ellipse is taller than it is wide. The larger denominator, a^2, is under the y-term. Here, a > b. Relationship between a, b, and c c^2 = a^2 - b^2 This crucial formula connects the distance to a vertex (a), the distance to a co-vertex (b), and the distance to a focus (c), all measured from the center. Use it to find one value when the other two are known.