I. Equations of a Circle a. At the origin center= r= b. Standard from: center= r=


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1 11.: Circle & Ellipse I cn Write the eqution of circle given specific informtion Grph circle in coordinte plne. Grph n ellipse nd determine ll criticl informtion. Write the eqution of n ellipse from rel world sitution. I. Equtions of Circle. At the origin center= r= b. Stndrd from: center= r= c. Generl Eqution d. Note: The leding coefficients MUST be the sme to be circle. II. Key Concepts:. To grph, locte the center, then move r units right, left, up nd down from the center. b. Midpoint =(, ). You need this if you re given the endpoints of the dimeter. c. Distnce = You need this to find the rdius given the center nd point on the circle. d. Tngent Line Touches the circle in ONE point (point of tngency) nd is perpendiculr to the dimeter. Perpendiculr lines hve slopes tht re negtive reciprocls. III. Equtions of n Ellipse (the set of ll points P in plne such tht the SUM of the distnce between P nd two fixed points, clled foci, is constnt) A. Stndrd form: ( x h ) ( y k ) + = or ( x h ) ( y k ) b 1 b + = 1 1. Center point =. > b (they look like footbll nd in footbll, size mtters) 3. Vertices units from the center point long the mjor xis (in this cse lso the Mjor xis the one with the foci) 4. Covertices  units from the center point long the minor xis (smller in length Minor xis) 5. Foci (plurl for focus) units from the center point long the trnsverse xis. c = b The sign for the foci is lwys opposite the sign between the nd b. 6. If is under the x then it is footbll on the ground (horizontl) but if it is under the y, then it is footbll on the tee (verticl) c 7. Eccentricity The mount of roundness. e = Circles e = 0, Ellipses 0 < e < 1. B. Generl Form: x + cy + dx + ey + f = 0. There is not b (it is used for bxy term). In this form, you must use completing the squre to convert to stndrd.
2 II. How to grph A. Put in stndrd form B. Determine the center point (, ) C. From the center point, move units long the xis. These re your vertices ( h ±, k ) or ( h, k ± ) D. From the center point, move units long the xis. There re your covertices or E. From the center point, move units long the xis. These re your ( h c, k ) or ( h, k c) ± ±. Use. F. From the focl points move LR to the LR endpoints LR = III. Model Problems Guided Prctice 3x + 3y 6x + 1y = 1 b On Your Own x + y + 6x + y = 4 7 9x + 4y 36x 4y + 36 = 0 4x + y 48x 14y = 0 Determine the conic section: 8x + 1y 4x + y = 6 Determine the conic section: 4x + 4y + 8x 1y =
3 Write the eqution of the circle in stndrd form: rdius=4 center t (,5) Write the eqution of the circle in stndrd form: center t (,5) contining (5,1) x + y = 4 x ( y ) + 3 = 8 ( x ) ( y ) = 4 Grph ( y ) x + =
4 ( x ) ( y + 3) + = ( x + 4) 16 Grph ( y ) = 1 11.: Hyperbol I cn Grph hyperbol nd determine ll of the criticl informtion. Determine the eqution of hyperbol given some chrcteristics. I. Hyperbol. The set of points in plne whose difference of the distnce from two fixed point (foci) is positive constnt. b. The is lwys first (not necessrily bigger) nd is on the trnsverse xis. c. Formuls i. Trnsverse xis is horizontl: ii. Trnsverse xis is verticl iii. Asymptotes of hyperbol: Note: underneth the yterm could be or b but whichever it is, it is the top prt of the slope s frction. iv. c = Notice the chnge s compred to the ellipse v. Generl Form:
5 II. How to grph A. Put in stndrd form (must equl 1) B. Determine the center point (, ). Mke sure you put h first! C. Move units long the trnsverse xis. These re the D. Move units long the xis. E. Drw box round the endpoints of nd b, then connect the digonls. These re the. F. Stying between the symptotes, connect bck to only. F. Find the LR endpoints III. Model Problems Guided Prctice b LR = On Your Own 9x y 36x 6y + 18 = 0 4x 3y + 8x + 16 = 0 Find the stndrd form of the hyperbol with foci (1,) & (5,) nd vertices (0,) & (4,). Find the stndrd form of the hyperbol with foci (,6) & (,6) nd vertices (,3) & (,3). Find the stndrd form of the hyperbol with vertices (1, ) nd (3, ) nd symptotes of y=x nd y=x+4 Find the stndrd form of the hyperbol with vertices (3,0) nd (3,6) nd symptotes of y=x nd y=x+6
6 ( y ) ( x ) = Grph 9( y + 6) 4( x + ) = 36 ( x ) ( y ) + 1 = Grph ( y ) ( x 1) + = 1 4 1
7 11.: Prbols I cn Identify nd lbel ll prts of prbol. Convert ll generl form equtions to stndrd form I. Definitions. Prbol: Grph of qudrtic eqution to which set of points in plne tht re the sme distnce from given point. b. Vertex: Midpoint of the grph; turning point c. Focus: distnce from the vertex; locted inside the prbol d. Directrix: fixed line used to define its shpe; locted outside the prbol e. Axis of Symmetry: A line tht divides plne figure or grph into congruent reflected hlves. f. Ltus Rectum: A line segment through the foci of the shpe g. Eccentricity: rtio to describe the shpe of the conic, e=1 II. Horizontl Stndrd From:. Horizontl Stndrd From Eqution: b. Focus Point: c. Directrix: d. Axis of Symmetry: e. L.R. endpoints: f. Note: If p is + it opens right nd if p is it opens left III. Verticl Stndrd From:. Verticl Stndrd From Eqution: b. Focus Point: c. Directrix: d. Axis of Symmetry: e. L.R. endpoints: f. Note: If p is + it opens up nd if p is it opens down IV. All stndrd form equtions. Center: b. Length of L.R.: c. Eccentricity:
8 III. Model Problems Guided Prctice y x y = 0 On Your Own x x y = 0 y x y = 0 Find the stndrd form of the prbol where the xis of symmetry is t y=1, directrix is t x = nd the focus (4,1) Write in stndrd form the eqution of prbol with the vertex t the origin nd the focus is t (,0) Write in stndrd form the eqution of prbol with the directrix y=6 nd focus point (0,6). Write the Stndrd form eqution of prbol where the vertex is (7,3) nd focus point (,3). Find the stndrd form of the prbol with vertex (,1) nd the directrix is t x=1
9 y = 4x x = (1/ 0) y Vertex: P: Focus point: Directrix: AOS: L.R. : Vertex: P: Focus point: Directrix: AOS: L.R. : ( y + 1) = 8( x + 1) ( x 7) = 8( y ) Vertex: P: Focus point: Directrix: AOS: L.R. : Vertex: P: Focus point: Directrix: AOS: L.R. :
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