Definition of an Ellipse
The definition of an ellipse is a Grade 11 Algebra 2 concept from the conic sections unit in enVision Algebra 2. An ellipse is the set of all points in a plane for which the sum of the distances to two fixed points (foci) is constant. This geometric definition gives rise to the standard equation (x−h)²/a² + (y−k)²/b² = 1, where (h, k) is the center, a is the semi-major axis length, and b is the semi-minor axis length. Ellipses appear in planetary orbits, satellite dishes, and optics — the foci have physical significance in all these applications.
Key Concepts
An ellipse is the set of all points in a plane such that the sum of the distances from any point on the ellipse to two fixed points called foci is constant. This constant sum property creates the characteristic oval shape of an ellipse.
Common Questions
What is the definition of an ellipse?
An ellipse is the set of all points in a plane where the sum of the distances from the point to two fixed points (called foci) is constant. If F₁ and F₂ are the foci and P is any point on the ellipse, then PF₁ + PF₂ = constant.
What is the standard form equation of an ellipse?
The standard form is (x−h)²/a² + (y−k)²/b² = 1, where (h, k) is the center, a is the semi-major axis (distance from center to vertex along the longer axis), and b is the semi-minor axis. If a > b, the major axis is horizontal; if b > a, it is vertical.
What are the foci of an ellipse and how do you find them?
The foci lie on the major axis at distance c from the center, where c² = a² − b². For a horizontal ellipse centered at (h, k), the foci are at (h ± c, k). The foci are always inside the ellipse, between the vertices.
What is the difference between an ellipse and a circle?
A circle is a special ellipse where a = b, so both foci coincide at the center. For a general ellipse, a ≠ b, giving it a stretched oval shape. The ellipse equation becomes the circle equation x² + y² = r² when a = b = r and (h, k) = (0, 0).
Where do ellipses appear in the real world?
Planets orbit the Sun in elliptical paths with the Sun at one focus (Kepler's first law). Whispering galleries and lithotripsy machines use the reflective property that sound or energy from one focus reflects to the other. Satellite dish cross-sections are often elliptical.
When do students learn the definition of an ellipse?
Ellipses are introduced in Grade 11 Algebra 2 as part of the conic sections unit. They build on completing-the-square skills and the distance formula.
Which textbook covers the definition of an ellipse?
The definition and equation of an ellipse are in enVision Algebra 2, used in Grade 11 math, in the conic sections chapter alongside circles, parabolas, and hyperbolas.