Parabolas Section 11.1

Transcription

1 Conic Sections Parabolas Section 11.1 Verte=(, ) Verte=(, ) Verte=(, ) 1 3 If the equation is =, then the graph opens in the direction. If the equation is =, then the graph opens in the direction. Parabola--- The set of all points in a plane equidistant from a fied point and a line in the plane. Focus-- the fied point. = a or = a If = a, then the focus is c units up from the verte. (0,a) If = a, then the focus is c units to the right of the verte. (a,0) Directri--the line If = a, then the directri is = -a If = a, then the directri is = - a 1

2 Find the Focus and the equation for the directri for the following and graph the parabola, focus, and directri. = a or = a = = 8 = 6 focus: directri: Equation for graphing: 1 8 focus: directri: Equation for graphing:

3 Find the equation of a parabola with verte at the origin, ais of smmetr the or ais, and: a) Directri = b) Directri = - c) Focus (0,5) d) Focus (-,0) e) -ais smmetr and contains the point (,8) f) -ais smmetr and contains the point (-5,10) g) A parabolic bridge is used with m son s Thomas the Trains set. The arch of the bridge is inches high and spans 6 inches. Find the equation of the arch. Hint: ou need to find a. The origin is placed at the verte of the parabola = a Find the points of intersection in the first quadrant for each pair of parabolas to three decimal places using our calculator. 3, 3 3

4 CIRCLE Section 11. EQUATION OF A CIRCLE: (,) represents an point on the circle. What we are looking for is an equation for the circle. If the distance between the points (, ) and ( 0, 0 ) is 3, then find and. r 0,0= r = 1. Find the equation of the circle with : a) a radius of and graph it. b) a radius of 10 and graph it. Radius= 1 The number under the give ou the units in the -direction from the center. The number under the give ou the units in the -direction from the center.

5 ELLIPSE In eercises 1 8, graph the given ellipse. 1) 1 ) =1 3) 1 ) ) ) 5 5 7) )

6 Definition: An ellipse is the set of all points in a plane such that the sum of their distances from two fied a b points is constant. The two fied points are called FOCI (the plural of the word FOCUS). 1 Length of one part of the string is a+c If we center the pencil on B, Length of the other part of the string is a-c Then we can form a right triangle So the length of the string is a Foci is c and - c units form the center along the major ais. If X is the major ais, then If is the major ais, then b c a a c b Where c a b if a>b or c b a if b>a Graph the ellipse, then find the vertices, major ais, and the foci of the given ellipse = 00 Vertices: Major ais: Foci: 6

7 Find the equation of the ellipse given the following information. a) major ais on -ais, major ais length =10, minor ais length = 6 b) major ais on -ais, major ais length = 8, minor ais length = c) major ais on -ais, major ais length =, distance of foci from center is 10 d) Find the equation where each of the points have a distance from (0,1) that is one-half of the distance from the line =. We cannot assume it is an ellipse. 7

8 Hperbolas Section 11.3 Definition: A hperbola is the set of all points in a plane such that the absolute difference of their distances from two fied points is constant. The two fied points are called FOCI. Standard Equation of a Horizontal Hperbola 1 -intercepts Vertices Vertices at a,0 and ( a,0) -intercepts: None Foci at a b c,0 and ( c,0) where c a b b Equations of asmptote lines: a Standard Equation of a Vertical Hperbola -intercepts: None b a 1 -intercepts Vertices Vertices at 0, b) and (0, b) c c c a b Foci at 0, and (0, ) where b Equations of asmptote lines: a *****The Vertices and Foci are on the same ais. *****A bo's corners are used to create the asmptotes *****The Foci are c units from the origin. *****The letter that comes first (positive coefficient) is the ais that the vertices and foci. *****The letter that comes first (positive coefficient) is the direction the graph opens. 1 Graph and find the foci and the equations of the asmptotes. 8

9 Graph and find the foci and the equations of the asmptotes. 1) 1 ) =1 3) 1 ) ) ) 5 5 7) ) 100

10 EQUATION OF A CIRCLE and MIDPOINT Section B-3 MIDPOINT = 1 1, M is the midpoint of A and B 1. A=(,3), B=(-3,0), M=. A=(,-3), B=(,), M= M is the midpoint of A and B 1. A=(0, 0), M=(, 3), B= B is at (, ). 1. B=(-5, 10), M= (5, 6), A= B is at (, ). 10

11 EQUATION OF A CIRCLE: (,) represents an point on the circle. What we are looking for is an equation for the circle. If the distance between the points (, ) and ( 1, ) is 3, then find and. h k r h, k= r = 1. Find the equation of the circle with : a) center at (-5,) and a radius of 3 b) center at (0,0) and a radius of 10 Find the center and the radius of the given circle. Graph the circle. ( 1) + ( 3) = ( + 3) + ( ) = Center: Radius: Center: Radius: 11

12 In eercises 17, find the center and the radius of the given circle. 17) ) ) ) 0 1) ) ) ) Center= Radius= 1

13 TRANSLATIONS (Just the find the equations, tpe of curve, and graph)(omit 65-77) Section 11. Graph. ( + 3) ( ) + = 1 5 ( + 3) ( ) = 1 5 Tpe of graph: Tpe of graph: Center= (, ) Center= (, ) ( 3) = ( + ) 13

14 1 Write the equation in standard form. Circle Ellipse Hperbola

15 Parabola = 0 + = ( + + ) = ( ) = ( ) = ( ) Tr: = 0 Finding the equation of the asmptotes in a hperbola. (+3) 5 ( ) = 1 ( k) = ± b ( h) a ( ) = ± ( ) Tr: ( + 7) 5 ( ) +=

16 Conics Review 1. Tell whether the graph of the ellipse has a major ais parallel with the or -ais. a) Vertices at (,3) and (, 5) b) Foci at (-,7) and (5, 7). Tell whether the graph of the hperbola opens in the or -direction. a) Vertices at (5,3) and (5, 7) b) Foci at (,-5) and (, -5) 3. Find the location of the foci for the ellipse, if c is 5, the center is (1,10), and one of the vertices is (1, 0). (1,10) (1, ). a) Find the center of the ellipse with foci at (-,7) and (5, 7). b) Find the center of the graph of the hperbola with foci (5,3) and (5, 7). 5. Find the equation and graph the ellipse with Vertices at (5,0) and (5, 10), and a focus at (5,) ( ) + ( ) = 1 Major Center Focus ( 5, 10 ) (, ) ( 5, ) (, ) (, ) (, ) b= c= 16

17 6. If the end points of the minor ais of an ellipse are at, 3 and 8, 3, and one of the two foci is at the point 5,1 ; find the equation of the ellipse. Graph the ellipse. ( ) + ( ) = 1 Minor Center Focus ( 8, 3 ) (, ) ( 5, 1 ) (, ) (, ) (, ) a= c= 7. Find the equation of the described parabola. Graph the parabola. Focus at the,1 and directri 3 Verte = 3 (,1) and (, 3) (, Focus ) (, 1 ) (, ) (, ) ( ) = a( ) a= 8. Find the equation of the hperbola with vertices is at 1, and 1, 1, 6. Write the equations of the asmptote lines and Graph the hperbola., and one of the two foci is at Opens in= Center Focus ( 1, ) (, ) ( 1, 6 ) (, ) (, ) (, ) b= c= 17

18 More of the same: 1. Find the equation and graph the ellipse with Vertices at (-10,) and (10, ), and a focus at (6,) ( ) + ( ) = 1 Major Center Focus ( 10, ) (, ) ( 6, ) (, ) (, ) (, ) a= c=. If the end points of the minor ais of an ellipse are at 3, and 3,6, and one of the two foci is at the point6, ; find the equation of the ellipse. Graph the ellipse. ( ) + ( ) = 1 Minor Center Focus ( 3, 6 ) (, ) ( 6, ) (, ) (, ) (, ) b= c= 18

19 3. Find the equation of the described parabola. Graph the parabola. Focus at the 5,3 and directri 1 ( Verte = 1 (5,3) and (1,3), Focus ) ( 5, 3 ) (, ) (, ) ( ) = a( ) a=. Find the equation of the hperbola with vertices is at 1,0 and 5,0 8,0. Write the equations of the asmptote lines and Graph the hperbola., and one of the two foci is at Opens in= Center Focus ( 5, 0 ) (, ) ( 8, 0 ) (, ) (, ) (, ) a= c= 1