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Introduction to Conics Conic Section: Circle Conic Section: Ellipse Conic Section: Parabola What are Conic Sections? 2 videos 1) Introduction to Conics (8 min.)Videos\Intro to Conics+Circles\Introduction to Conic Sections A.rv 2) Introduction to Circles (8:15 min.) Videos\Intro to Conics+Circles\Conic Sections Intro to Circles B.rv Circle is All points equidistant, r, from a single point, the center. (x,y) r • Standard “center radius” form of a Circle? (0,0) Center at (0,0) (x k)2 ( y k)2 r 2 Center at (h,k) (x,y) r ( x h) 2 ( y k ) 2 (h,k) 2 2 1 Examples follow r r 1) Center (0,0) and radius 3 (x-0)2+(y-0)2=9 x2+y2=9 32 2)Center (0,2) and radius 11 (x-0)2+(y-2)2 = 121 3)Center (3,4) and radius 2 (x-3)2+(y-4)2 = 4 4)Center (-4,0) and radius 5 (x+4)2+(y-0)2 = 25 5) Center (0,0) and radius 1/2 (x-0)2+(y-0)2 = 1/4 Circles a)Center (0,0) and radius 6 f) Center (3,0) and radius 9 b)Center (0,0) and radius 9 g) Center (0,–2) and radius 3 c) Center (0,0) and radius 11 h) Center ( 2, 3) and radius 6 d)Center (0,0) and radius 5 i) Center (–3, –5) and radius 5 e)Center (2,0) and radius 6 j) Center (–11, –12) and radius 4 What are Conic Sections? Video -- Introduction to Ellipse (13 min.) Videos\Intro to Ellipses C1.rv Definition: All points in a plane, the sum of whose distances from two fixed points (foci) is constant. The standard eq. form of an Ellipse ( x h) 2at (h, k)k ) 2 Center ( y 1 2 2 ( h, k+b) a b ( h–a, k) ( h+a, k) (h,k) ( h, k–b) ( 0, +3) 2 2 x y 1 ( –4, 0) ( 4, 0) 16 9 (0,0) a ( 0, –3) ( 0, +5) b x2 y2 1 ( x 3) 2 ( y 2) 2 16 25 1 25 16 ( –4, 0) ( 4, 0) c (0,0) ( 3, 6) ( –2, 0) ( 8, 0) ( 0, –5) (3,2) ( 3, –2) ( h, k+b) x2 y2 a 1 ( h–a, k) 36 25 ( h+a, k) (h,k) x2 ( h, k–b) b y2 1 9 ( h, k+a) c ( x 1) ( y 2) 2 2 1 16 25 (h,k) ( h–b, k) ( h+b, k) d ( x 2) 2 ( y 3) 2 1 9 4 ( h, k–a) Page 364, #s 35, 36, 39, 40, 41. 42 x2 y2 1 1 25 16 x2 y2 4 ( x 4) 2 ( y 5) 2 2 1 1 144 169 28 64 ( x 4) 2 y 2 5 3 1 4 x 9 y 2 36 9 5 An Ellipse has 2 foci Definition (reworded): an Ellipse is the set of points where the sum of the points’ distances from the 2 foci is a constant. Determining the location of the 2 foci… ..\7th 5 weeks\Foci of an Ellipse C2.rv Important relationships: Let the focus length be equal to c c2=a2-b2 d1 d2 d1+d2=2a Eccentricity (flatness), e = c/a, a c Examples follow What is the ellipse’s equation (in standard form) given… (-5,7) (-3,7) (3,7) (5,7) Vertices: (±5,7) Foci: (±3,7) c2=a2-b2 a c Since, a=5 & c=3, then b=4 ( x) 2 ( y ) 2 ( x) 2 ( y ) 2 2 2 1 1 a b 25 16 Note: The Foci are always on the major axis !! Sketching the ellipse first, might HELP ! Vertices: (±13,1) Foci: (±12,1) Vertices: (±4,7) Foci: (±3,7) Vertices : (2,1), (+14,1) Foci: (4,1), (+12,1) b Vertices: (7,±5) Foci: (7,±3) a c Page 364, #s 47 through 50 and 51 for extra credit √ √ √ √ √ How to quickly Identify the conic from the equation (future) ? (x,y) Circle: ( x h) 2 ( y k) 2 r 2 2 1 r r (h,k) ( x h) 2 ( y k ) 2 ( h, k+b) Ellipse: 1 a 2 b 2 ( h–a, k) d1 d2 ( h+a, k) c2=a2-b2 d1+d2=2a a c ( h, k–b) Eccentricity (flatness), e = c/a Parabola: y a ( x h) 2 k We have studied if vertex is at (0,0) y a(x) 2 parabolas that point up or down (so far). if vertex is at ( h, k) ( y k ) a ( x h) 2 Circle – set of all points that are the same distance (equidistant), r, from a single point, the center. Ellipse – set of all points in a plane, the sum of whose distances from two fixed points (foci) is constant. Parabola – set of points in a plane that are equidistant from a fixed line (the directrix) and a point (the focus). Parabola – opening up or down, the equation is: Point 1: And if the vertex (h,k) is at (0,0), then y a( x h) 2 k becomes x 2 4 py Similarly if we have a parabola opening left or right then the x and y is switched around y 2 4 px Point 2: p is the distance from the vertex to the focus and to the directrix Note By the definition of a parabola the vertex is always midway between the focus and the directrix. Point 3: Hence, to find that distance divide the coefficient of the variable (the variable having a 1 as its exponent) by 4. Reference Drawn examples on board SUMMARIZING… Remember the vertex is at ( 0, 0 )… if the parabola opens ‘up’ then the focus is at ( 0, p) if the parabola opens ‘down’ then the focus is at ( 0, -p) if the parabola opens to the ‘right’ then the focus is at ( p, 0) if the parabola opens to the ‘left’ then the focus is at ( -p, 0) x 2 ( A) y (4 p) y x 2 16 y set 4p=16 and solve for ‘p’ Opens up solved… p=4 therefore the focus is at ( 0, 4) 1 set 4p= –1/2 and solve for ‘p’ x y 2 2 solved… p= –1/8 Opens down therefore the focus is at ( 0, –1/8) y 2 9x set 4p=9 and solve for ‘p’ solved… p = 9/4 = 2 ¼ Opens right therefore the focus is at ( 2 ¼, 0) Page # 363, problem #s 1, 2, 3, 4, 11,12 Due Wednesday Page #363, problem #s 13, 14, 15, 16 Due TBD Page #363, problem #s 17, 18, 19, 20, 21, 22, 23 Due TBD x 2 16 y set 4p=16 and solve for ‘p’ Opens up the focus is at ( 0, 4) -- see previous slide and the directrix, y = –4 1 set 4p= –1/2 and solve for ‘p’ x y 2 2 the focus is at ( 0, –1/8) Opens down and the directrix, y = +1/8 y 2 9x set 4p=9 and solve for ‘p’ the focus is at ( 2 ¼, 0) Opens right and the directrix, x = –2 ¼ Eccentricity = e = c/a Explain what the effect is on the ellipse’s shape as the focus’s distance from the center (‘c’) approaches the vertex’s distance from the center (‘a’) -- in other words, when ‘e’ approaches a value of 1. Please note that the next school-wide writing prompt will take place on Tuesday, 4/5/11 during 2nd period. The prompt is as follows: "The use of Cornell Notes, Flash Cards and Concept Maps are currently used to help you organize your notes and make your test preparation easier. What other learning activities would you like to see incorporated in your class?" After the essays have been completed, please compile or ask a student to make a list of the ideas submitted by your class. Give this list to your ILT representative by the end of the day on 4/5/11. This will help the entire school!

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