1. An ellipse is the figure consisting of all those points for which the sum of their distances to two fixed points (called the foci) is a constant.
2. An ellipse is the figure consisting of all points in the plane whose Cartesian coordinates satisfy the equation
Where h, k, a and b are real numbers. a and b are positive.
An ellipse centered at the point (h, k) and having its major axis parallel to the x-axis may be specified by the equation
Parametric equations of the ellipse:
Major axis = 2a
Minor axis = 2b
Define a new constant called the eccentricity ( is the case of a circle) The eccentricity is:
The greater the eccentricity is, the more elongated is the ellipse.
If c equals the distance from the center to either focus, then
The distance between the foci is
The area enclosed by an ellipse is , where In the case of a circle where a = b, the expression reduces to the familiar
Say is a fixed point of the ellipse.
The equation of the tangent line in point of an ellipse
Given the following equation
a Find the length of the major and minor axes.
b) Find the coordinates of the foci.
c) Sketch the graph of the equation.
a) We first write the given equation in standard form:
The major axis length is given by = 2a = 4
The minor axis length is given by = 2b = 6
Sketch the graph of the ellipse whose equation is
We see that the center of the ellipse is (h, k) = (2, -1). Next, note that a = 3, b = 2.
We know that the endpoints of the major axis are exactly 3 units left and right the center, which places them at the points (-1, -1) and (5, -1).
We also know that the endpoints of the minor axis are exactly 2 units above and below the center, which places them at the points (2, 1)and (2, -3).