6.2 Ellipses and Hyperbolas

1. 6.2 Ellipses and Hyperbolas An ellipse is the set of all points in a plane the sum of whose distances from two fixed points is constant. Each fixed point is called a focus of the ellipse. • The graph of an ellipse is not that of a function. • The foci lie on the major axis – the line from V to V. • The minor axis – B to B

2. 6.2 The Equation of an Ellipse • Let the foci of an ellipse be at the points (c, 0). The sum of the distances from the foci to a point (x, y) on the ellipse is 2a. So, we rewrite the following equation.

3. 6.2 The Equation of an Ellipse

4. 6.2 The Equation of an Ellipse • Replacing a2– c2 with b2 gives the standard equation of an ellipse with the foci on the x-axis. • Similarly, if the foci were on the y-axis, we get

5. 6.2 The Equation of an Ellipse The ellipse with center at the origin and equation has vertices (a, 0), endpoints of the minor axis (0, b), and foci (c, 0), where c2 = a2– b2. The ellipse with center at the origin and equation has vertices (0, a), endpoints of the minor axis (b, 0), and foci (0, c), where c2 = a2– b2.

6. 6.2 Graphing an Ellipse Centered at the Origin Example Graph Solution Divide both sides by 36. This ellipse, centered at the origin, has x-intercepts 3 and –3, and y-intercepts 2 and –2. The domain is [–3, 3]. The range is [–2, 2].

7. 6.2 Finding Foci of an Ellipse Example Find the coordinates of the foci of the equation Solution From the previous example, the equation of the ellipse in standard form is Since 9 > 4, a2 = 9 and b2 = 4. The major axis is along the x-axis, so the foci have coordinates

8. 6.2 Finding the Equation of an Ellipse Example Find the equation of the ellipse having center at the origin, foci at (0, ±3), and major axis of length 8 units. Give the domain and range. Solution 2a = 8, so a = 4. Foci lie on the y-axis, so the larger intercept, a, is used to find the denominator for y2. The standard form is with domain and range [–4, 4].

9. 6.2 Ellipse Centered at (h, k) An ellipse centered at (h, k) with horizontal major axis of length 2a and vertical minor axis of length 2b has equation Similarly for ellipses with vertical major axis. Example Graph Analytic Solution Center at (2, –1). Since a > b, a = 4 is associated with the y2 term, so the vertices are on the vertical line through (2, –1).

10. 6.2 Ellipse Centered at (h, k) Graphical Solution Solving for y in the equation yields The + sign indicates the upper half of the ellipse, while the – sign yields the bottom half.

11. 6.2 Hyperbolas A hyperbola is the set of all points in a plane the difference of whose distances from two fixed points is constant. The two fixed points are called the foci of the hyperbola. • If the center is at the origin, the foci are at (±c, 0). • The midpoint of the line segment FF is the center of the hyperbola. • The vertices are at (±a, 0). • The line segment VV is called thetransverseaxis.

12. 6.2 Standard Form of Equations for Hyperbolas The hyperbola with center at the origin and equation has vertices (±a, 0) and foci (±c, 0), where c2 = a2 + b2. The hyperbola with center at the origin and equation has vertices (0, ±a) and foci (0, ±c), where c2 = a2 + b2. • Solving for y in the first equation gives If |x| is large, the difference approaches x2. Thus, the hyperbola has asymptotes

13. 6.2 Using Asymptotes to Graph a Hyperbola Example Sketch the asymptotes and graph the hyperbola Solutiona = 5 and b = 7 Choosing x = 5 (or –5) gives y = ±7. These four points: (5, 7), (–5, 7), (5, –7), and (–5, –7), are the corners of the fundamental rectangle shown. The x-intercepts are ±5.

14. 6.2 Graphing a Hyperbola with the Graphing Calculator Example Graph Solution Solve the given equation for y.

15. 6.2 Graphing a Hyperbola Translated from the Origin Example Graph Solution This hyperbola has the same graph as except that it is centered at (–3, –2).