Conic Sections Parabola Objective: Define conic section, parabola, draw a parabola, standard equations and their graphs

Transcription

1 Conic Sections Prol Ojective: Define conic section, prol, drw prol, stndrd equtions nd their grphs The curves creted y intersecting doule npped right circulr cone with plne re clled conic sections. If the plne cuts cler through one nppe nd is perpendiculr to the xis of the cone the conic is clled circle nd n ellipse if it s not perpendiculr. If the plne cuts only one nppe ut does not cut cler through the cone then the conic is prol. If the plne cut through oth nppes, ut not through the vertex the conic is clled hyperol. Distnce is involved in ll the definitions of conics so we need the distnce formul: Given two points (x 1, y 1 ) nd (x, y ): d = ( x x ) + ( y y ) 1 1 Ex: Find the distnce etween points (3,4) nd (-,5). Defintion of Prol: A prol is the set of ll points P(x, y) in plne whose distnce to fixed point, clled the focus, equls its distnce to fixed line, clled the directrix. The line through the focus perpendiculr to the directrix is clled the xis of symmetry, nd the point on the xis of symmetry hlfwy etween the directrix nd the focus is clled the vertex.

2 Equtions of prol with vertex (,) y = 4x x = 4y Vertex: (,) Vertex: (,) Focus: (,) Focus: (,) Directrix: x = - Directrix: y = - Symmetric with respect to x-xis Symmetric with respect to y-xis Axis of symmetry the x-xis Axis of symmetry the y-xis Ex: Locte the focus nd directrix nd sketch the grph of y = 4x Ex: Locte the focus nd directrix nd sketch the grph of x =3y Ex:. Find the eqution of the prol hving the origin s its vertex, y-xis s its xis of symmetry, nd (-1,-5) on its grph.. Find the coordintes of its focus nd the eqution of its directrix. Ex:. Find the eqution of the prol hving the origin s its vertex, y-xis s its xis of symmetry, nd (4,) on its grph.. Find the coordintes of its focus nd the eqution of its directrix. Circles nd Ellipse Ojective: Definition of circle nd Ellipse, Drw circle nd ellipse stndrd equtions Definiton of circle: The set of ll points P in plne tht re equidistnt from fixed point clled the center. The fixed distnce is the rdius. Eqution of circle centered t (,) An eqution for the circle with its center t (, ) nd rdius of r is x + y = r Ex: Grph nd find the stndrd form of the eqution of the circle with center t (, ) nd rdius 4. Ex: Find the center nd rdius of the circle with eqution x + y = 1

3 Definition of n ellipse: An ellipse is the set of ll points P in plne such tht the SUM of the distnces from P to two fixed points, F 1 nd F, (foci), is constnt. The line through the foci intersects the ellipse t two points clled the vertices.. The chord joining the vertices is the mjor xis, nd the midpoint is the center of the ellipse. The chord perpendiculr to the mjor xis is the minor xis. Equtions of n ellipse with center (,): An eqution for n ellipse with center (,), mjor xis length is, nd minor xis length is where < <, is Horizontl Mjor Axis: Verticl Mjor Axis: x y In ech cse: - = c ( ± c= foci) x y THE MAJOR AXIS IS ALWAYS LONGER THAN THE MINOR AXIS Ex: Grph nd find the stndrd form of the eqution of the ellipse centered t the origin with horizontl mjor xis = 14 nd minor xis = 1. Ex: Find the coordintes of the foci, find the lengths of the minor nd mjor xis nd grph: x y 4 9 Ex: Find the coordintes of the foci, find the lengths of the minor nd mjor xis nd grph: 16x + 9y = 144

4 Hyperol Ojective: Definition of hyperol, drw hyperol, eqution of hyperol Definition of Hyperol: A hyperol is the set of points P(x, y) in plne such tht the ABSOLUTE VALUE of the DIFFERENCE etween the distnces from P to two fixed points, F 1 nd F, clled the foci, is constnt. The intersection points of the line through the foci nd the two rnches of the hyperol re clled vertices, ech is clled vertex. The segment etween the two vertices is clled the trnsverse xis. The midpoint of the trnsverse xis is the center. The line perpendiculr to the trnsverse xis through the center is the conjugte xis. Eqution of hyperol with center (,): Horizontl Trnsverse Axis: Verticl Trnsverse Axis: x y = 1 y x = 1 Asymptotes y=± x Asymptotes y=± x In ech cse: + = c ( ± = vertices, ± c= foci) The length of the TRANSVERSE xis is The length of the CONJUGATE xis is. Ex: Find the coordintes of the foci, find the lengths of the trnsverse nd conjugte xes, find the equtions of the symptotes, nd grph the following eqution: x 9y = 9 Ex: Find the coordintes of the foci, find the lengths of the trnsverse nd conjugte xes, find the equtions of the symptotes, nd grph the following eqution: 4y 5x = 1 Ex: Find the eqution of the hyperol with verticl trnsverse xis centered t the origin with length of the trnsverse xis of 16, length of the conjugte xis is 4.

5 Conic Sections NOT Centered t the Origin Ojective: Identify Conics Identifying Conics The grph of n eqution of the form Ax + Bxy+ Cy + Dx+ Ey+ F = is 1. A hyperol if B 4AC >. A prol if B 4AC = 3. An ellipse if B 4AC< Stndrd equtions of prol with vertex (h, k): Verticl Prol: Horizontl Prol : 4(y k) = (x h) 4(x h) = (y k) > : opens upwrd > : opens right < : opens downwrd < : opens left focus: (h, k + ) focus: (h +, k) horizontl directrix: y = k verticl directrix: x = h Verticl xis of symmetry: x = h Horizontlxis of symmetry: y = k Ex: Find the focus nd directrix of 16( x 3) = ( y+ 5) Ex: Find the vertex, focus, nd directrix of y y x =. (you will need to know how to complete the squre) Stndrd eqution of circle with center (h,k): An eqution for the circle with its center t (h, k) nd rdius of r is ( ) ( ) x - h + y - k = r Ex: Grph nd find the stndrd form of the eqution of the circle with center t (, -5) nd rdius 4. Ex: Find the center nd rdius of the circle with eqution x + y 1x+ 6y+ 3=

6 Stndrd equtions of n ellipse (h, k): An eqution for n ellipse with center (h,k), mjor xis length is, nd minor xis length is where < <, is Horizontl Mjor Axis: Verticl Mjor Axis: - = c ( x - h) ( y - k) ( x - h) ( y - k) Ex: Find the center, vertices, nd foci of the ellipse with eqution 4x + y 8x+ 4y 8= Ex: Identify the conic x x y y = Stndrd eqution of hyperol with center (h, k): Horizontl Trnverse Axis: Verticl Trnsverse Axis: ( x h) ( y k) = 1 + = c The length of the TRANSVERSE xis is The length of the CONJUGATE xis is. ( y k) ( x h) = 1 Ex: Find the eqution of the hyperol with vertices on the line x = - 4, conjugte xis on the line y = 3, length of the trnsverse xis = 4, nd length of the conjugte xis = 6. Ex: Identify the conic 16x 5y 16x=