Section 13.2 The Ellipse

Objectives Define an ellipse Graph ellipses centered at the origin Graph ellipses centered at (h, k) Solve application problems involving ellipses

Objective 1: Define an Ellipse Definition of an Ellipse: An ellipse is the set of all points in a plane for which the sum of the distances from two fixed points is a constant. The figure below illustrates that any point on an ellipse is a constant distance d1 + d2 from two fixed points, each of which is called a focus. Midway between the foci is the center of the ellipse.

Objective 2: Graph Ellipses Centered at the Origin The Equation of an Ellipse Centered at the Origin: The standard form of the equation of an ellipse that is symmetric with respect to both axes and centered at (0, 0) is To graph an ellipse centered at the origin, it is helpful to know the intercepts of the graph. The graph of is an ellipse, centered at the origin, with x-intercepts (a, 0) and (–a, 0) and y-intercepts (0, b) and (0, –b).

Objective 2: Graph Ellipses Centered at the Origin For , if a > b, the ellipse is horizontal (figure (a)). If b > a, the ellipse is vertical (figure (b)). The points V1 and V2 are called the vertices of the ellipse. The line segment joining the vertices is called the major axis, and its midpoint is called the center of the ellipse. The line segment whose endpoints are on the ellipse and that is perpendicular to the major axis at the center is called the minor axis of the ellipse.

EXAMPLE 1 Graph: Strategy This equation is in standard form. We will identify a and b. Why Once we know a and b, we can determine the intercepts of the graph of the ellipse.

EXAMPLE 1 Solution Graph: The color highlighting shows how to compare the given equation to the standard form to find a and b. The x-intercepts are (a, 0) and (–a, 0), or (6, 0) and (–6, 0). The y-intercepts are (0, b) and (0, –b), or (0, 3) and (0, –3). Using these four points as a guide, we draw an oval-shaped curve through them, as shown in figure (a). The result is a horizontal ellipse because a > b.

EXAMPLE 1 Solution Graph: To increase the accuracy of the graph, we can find additional ordered pairs that satisfy the equation and plot them. For example, if x = 2, we have

EXAMPLE 1 Solution Graph: In a similar way, we can find the corresponding values of y for the x-value 4. In figure (b) we record these ordered pairs in a table, plot them, use symmetry with respect to the y-axis to plot four other points, and draw the graph of the ellipse.

Objective 3: Graph Ellipses Centered at (h, k) Not all ellipses are centered at the origin. As with the graphs of circles and parabolas, the graph of an ellipse can be translated horizontally and vertically. The Equation of an Ellipses Centered at (h, k): The standard form of the equation of a horizontal or vertical ellipse centered at (h, k) is For a horizontal ellipse, a is the distance from the center to a vertex. For a vertical ellipse, b is the distance from the center to a vertex.

EXAMPLE 3 Graph: Strategy The equation is in standard form. We will identify h, k, a, and b. Why If we know h, k, a, and b we can graph the ellipse.

EXAMPLE 3 Solution Graph: To determine h, k, a, and b, we write the equation in the form We find the center of the ellipse in the same way we would find the center of a circle, by examining (x – 2)2 and (y + 3)2. Since h = 2 and k = –3, this is the equation of an ellipse centered at (h, k) = (2, –3). From the denominators, 42 and 52, we find that a = 4 and b = 5. Because b > a, it is a vertical ellipse. We first plot the center. Since b is the distance from the center to a vertex for a vertical ellipse, we can locate the vertices by counting 5 units above and 5 units below the center. The vertices are the points (2, 2) and (2, –8). To locate two more points on the ellipse, we use the fact that a is 4 and count 4 units to the left and to the right of the center. We see that the points (–2, –3) and (6, –3) are also on the graph. Using these four points as guides, we draw the graph as shown:

Objective 4: Solve Application Problems Involving Ellipses Ellipses, like parabolas, have reflective properties that are used in many practical applications. For example, any light or sound originating at one focus of an ellipse is reflected by the interior of the figure to the other focus.

EXAMPLE 5 Landscape Design. A landscape architect is designing an elliptical pool that will fit in the center of a 20-by-30-foot rectangular garden, leaving 5 feet of clearance on all sides, as shown in the illustration below. Find the equation of the ellipse. Strategy We will establish a coordinate system with its origin at the center of the garden. Then we will determine the x- and y-intercepts of the edge of the pool. Why If we know the x- and y-intercepts of the graph of the edge of the elliptical pool, we can use that information to write its equation.

EXAMPLE 5 Landscape Design. A landscape architect is designing an elliptical pool that will fit in the center of a 20-by-30-foot rectangular garden, leaving 5 feet of clearance on all sides, as shown in the illustration below. Find the equation of the ellipse. Solution We place the rectangular garden in the coordinate system shown below. To maintain 5 feet of clearance at the ends of the ellipse, the x-intercepts must be the points (10, 0) and (–10, 0). Similarly, the y-intercepts are the points (0, 5) and (0, –5). Since the ellipse is centered at the origin, its equation has the form with a = 10 and b = 5. Thus, the equation of the boundary of the pool is