6.3: Ellipses - Honors Precalculus Comprehensive Study Guide

Objectives for Chapter 6.3: Ellipses

At the end of this lesson on ellipses, students must be able to perform the following tasks:

• Reduce the general equation of an ellipse to its standard form and be able to graph the resulting ellipse.

• Find the specific equation of an ellipse that satisfies a given set of conditions.

Definition of an Ellipse

• An ellipse is formally defined as the set of all points in the plane, the sum of whose distances from two fixed points (known as the foci) is a constant value.

Standard and General Equations of the Ellipse

Standard Equations

The standard form of an ellipse depends on the orientation of its major axis (horizontal vs. vertical). The center of the ellipse is defined at the coordinates $(h, k)$.

• Horizontal Orientation (Major Axis parallel to the x-axis):   $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$

• Vertical Orientation (Major Axis parallel to the y-axis):   $\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1$

General Equation

The general equation of an ellipse is represented as a second-degree polynomial in two variables:

• $Ax^2 + By^2 + Dx + Ey + F = 0$

• This is a quadratic in two variables where the coefficients of the quadratic terms ($A$ and $B$) are not equal ($A \neq B$). Both coefficients must have the same sign (both positive or both negative) for the figure to be an ellipse.

Properties of the Ellipse

1. Center: Located at the point $(h, k)$.

2. Axes of Symmetry:

   • Major Axis: This is the longer axis. Its total length is defined as $2a$.

   • Minor Axis: This is the shorter axis. Its total length is defined as $2b$.

   • Core Relationship: The major axis is always longer than the minor axis ($a > b$).

3. Vertices: An ellipse has four distinct vertices.

   • Ends of Major Axis: These are denoted as $V_1$ and $V_2$.

   • Ends of Minor Axis: These are denoted as $V_3$ and $V_4$.

4. Foci:

   • There are two foci, denoted as $F_1$ and $F_2$.

   • The distance between the two foci is defined as $2c$.

5. Lateral Recta (Latus Rectum):

   • The distance from a focus to one endpoint of the latus rectum is $\frac{b^2}{a}$.

   • The total length of each latus rectum is $\frac{2b^2}{a}$.

6. Directrices:

   • The distance from the center of the ellipse to a directrix is $\frac{a}{e}$ (which can also be written as $\frac{a^2}{c}$).

   • The total distance between the two directrices is $\frac{2a}{e}$ (which can also be written as $\frac{2a^2}{c}$).

7. Pythagorean Relationship: (Note: The transcript lists this section as item 6 as well)

   • The relationship between the constants $a$, $b$, and $c$ is given by: $a^2 = b^2 + c^2$.

   • In an ellipse, "a" is the longest segment, representing the length of the semi-major axis.

Eccentricity

Eccentricity ($e$) is a numerical parameter associated with conic sections that measures how much the conic section varies from being a perfect circle.

• Formula for Eccentricity:   $e = \frac{c}{a}$

• Eccentricity Values for Conic Sections:

  • Ellipse: $0 < e < 1$

  • Circle: $e = 0$

  • Parabola: $e = 1$

  • Hyperbola: $e > 1$

Exercise A: Reducing and Graphing Ellipses

The following equations must be reduced to standard form in order to identify features and graph the ellipse:

1. $\frac{x^2}{5} - \frac{y^2}{10} = -2$

2. $9x^2 + 4y^2 + 72x - 24y + 144 = 0$

3. $3x^2 + 4y^2 + 12x - 16y - 32 = 0$

4. $25(x - 2)^2 + 4(y + 5)^2 = 100$

Finding the Equation of an Ellipse

To successfully find the equation of a specific ellipse, three primary pieces of information are required:

1. The Center $(h, k)$.

2. The values of constant parameters $a$ and $b$.

3. The Orientation (whether the major axis is horizontal or vertical).

Exercise B: Determining Ellipse Equations from Given Conditions

Find the equation of the following ellipses based on the provided parameters:

1. Case 1:

   • Center: $(2, -3)$

   • Major axis: Parallel to the x-axis

   • One vertex: Located at $(2, -1)$

   • One focus: Located at $(5, -3)$

2. Case 2:

   • Center: $(2, -1)$

   • Vertex: $(2, \frac{1}{2})$

   • Minor axis: Total length of $2$

3. Case 3:

   • Eccentricity ($e$): $\frac{3}{5}$

   • Semimajor axis ($a$): $5$

   • Center: At the origin $(0, 0)$

4. Case 4:

   • Latus rectum: $\sqrt{3}$

   • Major axis: Parallel to the x-axis

   • Center: At the origin $(0, 0)$

   • Passing through the point: $(2, -\sqrt{2})$