2 Objectives Write the standard equation for an ellipse. Graph an ellipse, and identify its center, vertices, co-vertices, and foci.
3 EllipseIf you pulled the center of a circle apart into two points, it would stretch the circle into an ellipse.An ellipse is the set of points P(x, y) in a plane such that the sum of the distances from any point P on the ellipse to two fixed points F1 and F2, called the foci (singular: focus), is the constant sum d = PF1 + PF2. This distance d can be represented by the length of a piece of string connecting two pushpins located at the foci.You can use the distance formula to find the constant sum of an ellipse.
4 Definition of constant sum It is the sum of the distances from the focus to a point on the ellipse
5 Example 1: Using the Distance Formula to Find the Constant Sum of an Ellipse Find the constant sum for an ellipse with foci F1 (3, 0) and F2 (24, 0) and the point on the ellipse (9, 8).Solution:d = PF1 + PF2 Definition of the constant sum of an ellipseDistance Formulad = 27The constant sum is 27
6 ExampleFind the constant sum for an ellipse with foci F1 (0, –8) and F2 (0, 8) and the point on the ellipse (0, 10).Solution:The constant sum is 20.
7 ELLIPSESInstead of a single radius, an ellipse has two axes. The longer the axis of an ellipse is the major axis and passes through both foci. The endpoints of the major axis are the vertices of the ellipse. The shorter axis of an ellipse is the minor axis. The endpoints of the minor axis are the co-vertices of the ellipse. The major axis and minor axis are perpendicular and intersect at the center of the ellipse.
8 ELLIPSESThe standard form of an ellipse centered at (0, 0) depends on whether the major axis is horizontal or vertical.
9 ELLIPSESThe values a, b, and c are related by the equation c2 = a2 – b2. Also note that the length of the major axis is 2a, the length of the minor axis is 2b, and a > b.
10 Example 2A: Using Standard Form to Write an Equation for an Ellipse Write an equation in standard form for each ellipse with center (0, 0).Vertex at (6, 0); co-vertex at (0, 4)SOLUTION:Step 1 Choose the appropriate form of equation.x2a2= 1y2b2The vertex is on the x-axis.
11 SOLUTION x2 y2 + = 1 36 16 Step 2 Identify the values of a and b. a = The vertex (6, 0) gives the value of a.b = 4 The co-vertex (0, 4) gives the value of b.Step 3 Write the equation.x236= 1y216
12 EXAMPLEWrite an equation in standard form for each ellipse with center (0, 0).Co-vertex at (5, 0); focus at (0, 3)Solution:Step 1 Choose the appropriate form of equation.y2a2= 1x2b2The vertex is on the y-axis.Step 2 Identify the values of b and c.b = 5The co-vertex (5, 0) gives the value of b.c = 3The focus (0, 3) gives the value of c.
13 solution Step 3 Use the relationship c2 = a2 – b2 to find a2. y2 x2 32 = a2 – Substitute 3 for c and 5 for b.a2 = 34Step 4 Write the equationy234= 1x225
14 exampleWrite an equation in standard form for each ellipse with center (0, 0).Vertex at (9, 0); co-vertex at (0, 5)
15 Ellipses with a center not in the origin Ellipses may also be translated so that the center is not the origin.
16 Example 3: Graphing Ellipses Graph the ellipseSolution:Step 1 Rewrite the equation asStep 2 Identify the values of h, k, a, and bh = –4 and k = 3, so the center is (–4, 3).a = 7 and b = 4; Because 7 > 4, the major axis is horizontal.
17 solutionStep 3 The vertices are (–4 ± 7, 3) or (3, 3) and (–11, 3), and the co-vertices are (–4, 3 ± 4), or (–4, 7) and (–4, –1).
18 ExampleSolution:Step 1 Rewrite the equation asGraph the ellipse
19 solution Step 2 Identify the values of h, k, a, and b. h = 0 and k = 0, so the center is (0, 0).a = 8 and b = 5; Because 8 > 5, the major axis is horizontal.Step 3 The vertices are (±8, 0) or (8, 0) and (–8, 0), and the co-vertices are (0, ±5,), or (0, 5) and (0, –5).
20 Example 4: Engineering Application A city park in the form of an ellipse with equation , measured in meters, is being renovated. The new park will have a length and width double that of the original park.Solution:Find the dimensions of the new parkStep 1 Find the dimensions of the original park. Because 50 > 20, the major axis of the park is horizontal.a2 = 50, so and the length of the park isb2 = 20, so and the width of the park is
21 solution Step 2 Find the dimensions of the new park. The length of the park isThe width of the park isWrite an equation for the design of the new park.The equation in standard form for the new parkwill be.
22 Student guided practice Do even problems from 2-10 in your book page 834