Conic Sections. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics

Transcription

1 Conic Sections MATH 211, Calculus II J. Robert Buchanan Department o Mathematics Spring 2018

2 Introduction The conic sections include the parabola, the ellipse, and the hyperbola. y y y x x x

3 Parabola A parabola is the set o all points in the plane that are equidistant rom a ixed point called the ocus and a ixed line called the directrix. The vertex is the point on the parabola at the intersection o the line perpendicular to the directrix and passing through the ocus. y d v x

4 Example Find the equation o the parabola with ocus at (1, 2) and directrix y = y d x

5 Solution Let the point with coordinates (x, y) lie on the parabola, then by the deinition o the parabola, (x 1) 2 + (y 2) 2 = y ( 2) (x 1) 2 + (y 2) 2 = (y + 2) 2 (x 1) 2 + y 2 4y + 4 = y 2 + 4y + 4 (x 1) 2 4y = 4y 8y = (x 1) 2 y = 1 8 (x 1)2.

6 General Formula or the Parabola (1 o 2) Theorem The parabola with vertex at (b, c), ocus at (b, c + 1 4a ), and directrix given by y = c 1 4a has equation y = a(x b)2 + c.

7 General Formula or the Parabola (1 o 2) Theorem The parabola with vertex at (b, c), ocus at (b, c + 1 ), and directrix 4a given by y = c 1 4a has equation y = a(x b)2 + c. ( y c + 1 4a ( (y c) + 1 4a (y c) (y c) + 2a ) 2 ( = (x b) 2 + y c 1 4a ) 2 ) 2 ( = (x b) 2 + (y c) 1 4a 16a 2 = (x b) 2 + (y c) 2 1 2a 1 (y c) a = (x b)2 y = a(x b) 2 + c ) 2 (y c) a 2

8 General Formula or the Parabola (2 o 2) Remark: i we switch the roles o x and y we have the ollowing result. Theorem The parabola with vertex at (c, b), ocus at (c + 1 4a, b), and directrix given by x = c 1 4a has equation x = a(y b)2 + c.

9 Example Given the equation or a parabola y = 2(x + 2) 2 + 1, ind the vertex, ocus, and directrix. 0-5 y x

10 Example Given the equation or a parabola y = 2(x + 2) 2 + 1, ind the vertex, ocus, and directrix. 0-5 y vertex = ( 2, 1), ocus = ( 2, 7/8), directrix: y = 9/8 x

11 Relective Property o Parabolas I a parabola is thought o as a relector (or example in a lashlight or satellite dish), all rays traveling perpendicular to the directrix and striking the parabola are relected through the ocus. d

12 Ellipse An ellipse is the set o all points in the plane or which the sum o the distances rom two ixed points called oci is a constant. y x

13 Example Suppose an ellipse has oci located at (1, 2) and (1, 4) and the point with coordinates (1, 1) lies on the ellipse. Find the equation o the ellipse. 5 4 y x

14 Solution (x 1) 2 + (y 2) 2 + (x 1) 2 + (y 4) 2 = K (1 1) 2 + (1 2) 2 + (1 1) 2 + (1 4) 2 = K = 4 (x 1) 2 + (y 2) 2 = 4 (x 1) 2 + (y 4) 2 (x 1) 2 + (y 2) 2 = 16 8 (x 1) 2 + (y 4) 2 (x 1) (x 1) 2 + (y 4) 2 (y 2) 2 = 16 8 (x 1) 2 + (y 4) 2 + (y 4) 2 y 7 = 2 (x 1) 2 + (y 4) 2 (y 7) 2 = 4(x 1) 2 + 4(y 4) 2 (y 3)2 4 = 1

15 General Formula or Ellipse Theorem The equation (x x 0) 2 a 2 + (y y 0) 2 b 2 = 1 with a > b > 0 describes an ellipse with oci at (x 0 c, y 0 ) and (x 0 + c, y 0 ) where c = a 2 b 2. The center o the ellipse is at the point (x 0, y 0 ) and the vertices are located at (x 0 ± a, y 0 ) on the major axis. The endpoints o the minor axis are at (x 0, y 0 ± b). Remark: the roles o the major and minor axes are reversed when b > a > 0.

16 Anatomy o an Ellipse minor axis v major axis c v

17 Example Identiy the ollowing eatures o the ellipse (x + 1) 2 (y 3)2 + = Center 2. Foci 3. Vertices 4. Endpoints o minor axis

18 Example Identiy the ollowing eatures o the ellipse (x + 1) 2 (y 3)2 + = Center ( 1, 3) 2. Foci ( 1 ± 5, 3) 3. Vertices ( 4, 3) and (2, 3) 4. Endpoints o minor axis ( 1, 1) and ( 1, 5)

19 Relective Property o Ellipses A ray emanating rom one ocus will always relect o the ellipse and pass through the other ocus.

20 Hyperbola Deinition A hyperbola is the set o all points in the plane or which the dierence o the distances rom two ixed points called oci is a constant. y x

21 Example Suppose a hyperbola has oci located at ( 2, 2) and (6, 2) and the point with coordinates (0, 2) lies on the hyperbola. Find the equation o the hyperbola. 6 4 y x

22 General Formula or the Hyperbola Theorem The equation (x x 0) 2 a 2 (y y 0) 2 b 2 = 1 describes a hyperbola with oci at (x 0 c, y 0 ) and (x 0 + c, y 0 ) where c = a 2 + b 2. The center o the hyperbola is at the point (x 0, y 0 ) and the vertices are located at (x 0 ± a, y 0 ). The asymptotes are the lines y = ± b a (x x 0) + y 0. Theorem The equation (y y 0) 2 a 2 (x x 0) 2 b 2 = 1 describes a hyperbola with oci at (x 0, y 0 c) and (x 0, y 0 + c) where c = a 2 + b 2. The center o the hyperbola is at the point (x 0, y 0 ) and the vertices are located at (x 0, y 0 ± a). The asymptotes are the lines y = ± a b (x x 0) + y 0.

23 Anatomy o a Hyperbola asymptote y v c v asymptote x

24 Example Identiy the ollowing eatures o the hyperbola x 2 (y 1)2 = Center 2. Foci 3. Vertices 4. Asymptotes

25 Example Identiy the ollowing eatures o the hyperbola x 2 (y 1)2 = Center (0, 1) 2. Foci (±2 5, 1) 3. Vertices (±2, 1) 4. Asymptotes y = ±2x + 1

26 Relective Property o Hyperbolas A ray directed toward one ocus will relect o the hyperbola and travel toward the other ocus.

27 Combination o Relecting Properties A relecting parabola and relecting hyperbola can be used together to ocus incoming rays on a single point. parabola hyperbola /v

28 Homework Read Section 10.5 Exercises: WebAssign/D2L