Math 180 Chapter 10 Lecture Notes. Professor Miguel Ornelas

Transcription

1 Math 180 Chapter 10 Lecture Notes Professor Miguel Ornelas 1

2 M. Ornelas Math 180 Lecture Notes Section 10.1 Section 10.1 Parabolas Definition of a Parabola A parabola is the set of all points in a plane equidistant from a fied point F (the focus) and a fied line l (the directri) that lie in the plane Parabolas with Verte V(0, 0) Equation, focus, directri Graph for p > 0 Graph for p < 0 2 = 4p or = 1 4p 2 Focus: F(0, p) Directri: = p 2 = 4p or = 1 4p 2 Focus: F(p, 0) Directri: = p Section 10.1 continued on net page... 2

3 M. Ornelas Math 180 Lecture Notes Section 10.1 (continued) Parabolas with Verte V(h, k) Equation, focus, directri Graph for p > 0 Graph for p < 0 ( h) 2 = 4p( k) or = a 2 + b + c, where p = 1 4a Focus: F(h, k + p) Directri: = k p ( k) 2 = 4p( h) or = a 2 + b + c, where p = 1 4a Focus: F(h + p, k) Directri: = h p Find the verte, focus, and directri of the parabola. Sketch its graph. 8 = = 0 Section 10.1 continued on net page... 3

4 M. Ornelas Math 180 Lecture Notes Section 10.1 (continued) Find an equation of the parabola with focus F( 4, 0) and directri = 4. Find an equation of the parabola with focus F(4, 2) and directri = 6. Section 10.2 Ellipses Definition of an Ellipse An ellipse is the set of all points in a plane, the sum of whose distances from two fied points (the foci) in the plane is a positive constant Section 10.2 continued on net page... 4

5 M. Ornelas Math 180 Lecture Notes Section 10.2 (continued) Standard Equations of an Ellipse with Center at the Origin The graph of 2 a b = 1 or 2 2 b a = 1 2, where a > b > 0, is an ellipse with center at the origin. The length of the major ais is 2a, and the length of the minor ais is 2b. The foci are a distance c from the origin, where c 2 = a 2 b 2. Find the vertices and foci of the ellipse. Sketch its graph = 1 ( + 2) ( 3)2 4 = 1 Section 10.2 continued on net page... 5

6 M. Ornelas Math 180 Lecture Notes Section 10.2 (continued) = 0 Find an equation of the ellipse with vertices V(0, ±7) and foci F(0, ±2). 6

7 M. Ornelas Math 180 Lecture Notes Section 10.3 Section 10.3 Hperbolas Definition of a Hperbola A hperbola is the set of all points in a plane, the difference of whose distances from two fied points (the foci) in the plane is a positive constant Standard Equations of a Hperbola with Center at the Origin The graph of 2 a 2 2 b 2 = 1 or 2 a 2 2 b 2 = 1, is a hperbola with center at the origin. The length of the transverse ais is 2a, and the length of the conjugate ais is 2b. The foci are a distance c from the origin, where c 2 = a 2 + b 2. Find the vertices, the foci, and the equations of the asmptotes of the hperbola. Sketch its graph = 1 Section 10.3 continued on net page... 7

8 M. Ornelas Math 180 Lecture Notes Section 10.3 (continued) ( + 2) 2 9 ( + 2)2 4 = = 0 Find an equation of the hperbola with vertices V(±5, 0) and foci F(±8, 0). Section 10.3 continued on net page... 8

9 M. Ornelas Math 180 Lecture Notes Section 10.3 (continued) Find an equation of the hperbola with vertices V(0, ±6) and asmptotes = ±3. Section 10.4 Plane Curves and Parametric Equations Definition of Plane Curve A plane curve is a set C of ordered pairs ( f (t), g(t)), where f and g are functions defined on an interval I. Definition of Parametric Equations Let C be the curve consisting of all ordered pairs ( f (t), g(t)), where f and g are functions defined on an interval I. The equations = f (t), for t in I, are parametric equations for C with parameter t. = g(t), Find an equation in and whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation. = t 2, = 2t + 3; 0 t 5 Section 10.4 continued on net page... 9

10 M. Ornelas Math 180 Lecture Notes Section 10.4 (continued) = t 3 + 1, = t 3 1; 2 t 2 = 2 3 sin t, = 1 3 cos t; 0 t 2π = e t, = e 2t ; t in IR 10

11 M. Ornelas Math 180 Lecture Notes Section 10.5 Section 10.5 Polar Coordinates Relationships Between Rectangular and Polar Coordinates The rectangular coordinates (, ) and polar coordinates (r, θ) of a point P are related as follows: 1. = r cos θ, = r sin θ 2. r 2 = 2 + 2, tan θ = if 0 If (r, θ) = (4, 7π/6) are polar coordinates of a point P, find the rectangular coordinates of P. If (, ) = ( 1, 3 ) are rectangular coordinates of a point P, find three different pairs of polar coordinates (r, θ) for P. Section 10.5 continued on net page... 11

12 M. Ornelas Math 180 Lecture Notes Section 10.5 (continued) Find a polar equation that has the same graph as the equation in and. a. = 3 b. ( + 2) = 16 Find an equation in and that has the same graph as the polar equation. Use it to help sketch the graph in rθ-plane. r = 5 r(sin θ 2 cos θ) = 6 Section 10.5 continued on net page... 12

13 M. Ornelas Math 180 Lecture Notes Section 10.5 (continued) r 2 (cos 2 θ + 4 sin 2 θ) = 16 Sketch the graph of the polar equation. θ = π 4 r = 6(1 + cos θ) Section 10.5 continued on net page... 13

14 M. Ornelas Math 180 Lecture Notes Section 10.5 (continued) Section 10.6 Polar Equations of Conics Theorem on Polar Equations of Conics A polar equation that has one of the four forms de de r = or r = 1 ± e cos θ 1 ± e sin θ is a conic section. The conic is a parabola if e = 1, an ellipse if 0 < e < 1, or a hperbola if e > 1. Find the eccentricit, and classif the conic. Sketch the graph, and label the vertices. r = sin θ r = 4 cos θ 2 Section 10.6 continued on net page... 14

15 M. Ornelas Math 180 Lecture Notes Section 10.6 (continued) Find an equation in and for the polar equation r = sin θ. Find a polar equation of the conic with focus at the pole that has the given eccentricit and equation of directri. a. e = 1, r = 2 sec θ b. e = 1, r sin θ = 2 3 Find a polar equation of the parabola with focus at the pole and verte V ( 4, π 2 ). 15