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Analytic Geometry: (lesson 2 of 3)

Ellipse

Definitions:

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

Ellipse Definitions

Where h, k, a and b are real numbers. a and b are positive.

Formulas:

Equations

An ellipse centered at the point (h, k) and having its major axis parallel to the x-axis may be specified by the equation

math example

Parametric equations of the ellipse:

math example

Major axis = 2a

Minor axis = 2b

Eccentricity

Define a new constant math examplecalled the eccentricity ( math solution is the case of a circle) The eccentricity is:

example.

The greater the eccentricity is, the more elongated is the ellipse.

Foci:

If c equals the distance from the center to either focus, then

math homework help

The distance between the foci is Ellipse foci

Area

The area enclosed by an ellipse is Ellipse area, where In the case of a circle where a = b, the expression reduces to the familiar analytic geometry

Tangent line

Say Ellipse tangent line is a fixed point of the ellipse.

The equation of the tangent line in point Ellipse tangent lineof an ellipse

hyperbola

Example 1:

Given the following equation

Ellipse example

a Find the length of the major and minor axes.

b) Find the coordinates of the foci.

c) Sketch the graph of the equation.

Solution:

a) We first write the given equation in standard form:

Ellipse example

The major axis length is given by = 2a = 4

The minor axis length is given by = 2b = 6

b)

Ellipse example

c)

Ellipse example

Example 2:

Sketch the graph of the ellipse whose equation is Ellipse example

Solution:

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).

Foci:

Ellipse solution

Ellipse solution