9.4
Ellipses
An ellipse forms when a right circular cone is sliced by an angled plane that doesn’t intersect its base, creating a closed curve.
Geometrically, an ellipse is the set of all points for which the sum of the distances to two fixed points—called foci—is constant.
The longest diameter is the major axis, and the shortest is the minor axis. These axes intersect at the center, and the major axis ends at the vertices.
The standard form is derived by centering the origin and placing the foci on the x-axis at -c and +c, where c is the distance from the center to each focus.
From any point on the ellipse, the total distance to both foci is the length of the major axis, expressed as a sum of two square roots. Rearranging and squaring both sides eliminates one square root. Squaring again removes all square roots, producing an equation with squared terms. Substituting the relationship between the axis lengths and focal distance, then rearranging, gives the standard form.
The larger denominator corresponds to the major axis, regardless of the ellipse’s center.
This equation describes real-world elliptical paths like planetary orbits and satellite motion.
An ellipse is formed when a right circular cone is intersected by an inclined plane that does not cut through its base. This intersection yields a closed, symmetric curve characterized by distinctive geometric properties. Most notably, an ellipse is defined as the collection of all points in a plane for which the combined distances to two fixed points—called the foci—remain constant.
The ellipse features two principal axes: the major and the minor axes. The major axis is the longest diameter, passing through both foci and the center of the ellipse. The minor axis is the shortest diameter and is oriented perpendicular to the major axis at the ellipse's center. For any point on the ellipse, the sum of the distances to the two foci is equal to the length of the major axis.
The general equation of an ellipse with its center at the origin (0,0) is:
Here, a and b represent the lengths of the semi-major and semi-minor axes, respectively, with a > b. If the center is at a point (h, k), the equation shifts to:
This canonical form is essential in modeling real-world elliptical trajectories, including the elliptical orbits described by Kepler's laws. It illustrates the deep connection between geometry and physics.
An ellipse forms when a right circular cone is sliced by an angled plane that doesn’t intersect its base, creating a closed curve.
Geometrically, an ellipse is the set of all points for which the sum of the distances to two fixed points—called foci—is constant.
The longest diameter is the major axis, and the shortest is the minor axis. These axes intersect at the center, and the major axis ends at the vertices.
The standard form is derived by centering the origin and placing the foci on the x-axis at -c and +c, where c is the distance from the center to each focus.
From any point on the ellipse, the total distance to both foci is the length of the major axis, expressed as a sum of two square roots. Rearranging and squaring both sides eliminates one square root. Squaring again removes all square roots, producing an equation with squared terms. Substituting the relationship between the axis lengths and focal distance, then rearranging, gives the standard form.
The larger denominator corresponds to the major axis, regardless of the ellipse’s center.
This equation describes real-world elliptical paths like planetary orbits and satellite motion.
From Chapter 9:
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9.4 : Ellipses
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9.2 : Parabolas
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9.5 : Eccentricity of an Ellipse
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9.6 : Hyperbolas
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9.7 : Geometry of Hyperbolas
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