Achievement Stndrd 909 Sketch grphs of conic sections nd write equtions relted to conic sections Clculus.5 Eternll ssessed credits Sketching Conics the Circle nd the Ellipse Grphs of the conic sections (circle, ellipse, prol nd hperol) nd their importnt fetures need to e known. The equtions of these grphs m e given in Crtesin form (n eqution linking nd, eg + = 1) or in prmetric form (two equtions defining ech of nd in terms of prmeter such s θ or t, eg = r cos θ, = r sin θ). The Circle The circle centre the origin, rdius hs Crtesin eqution + = + = nd prmetric equtions = cos θ, = sin θ If this circle is trnslted, the imge is circle, centre (, ), rdius, whose Crtesin eqution is ( ) + ( ) = nd whose prmetric equtions re = + cos θ = + sin θ To drw circles whose equtions re not in the stndrd form ove, rerrngement nd/or completing the squre m e required. Emple Q. Sketch the circle + + + = 0. ( ) + ( ) = A. ( ) + ( + ) = [grouping terms in nd terms in nd putting the constnt on the HS] ( + ) + ( + + 9) = + + 9 [completing the squre dding ( 1 the coefficient of ) ( ) + ( + ) = 9 nd ( 1 the coefficient of ) to ech side] which is circle, centre (, ), rdius 9 = whose grph is shown. Other fetures of interest re the intercepts. To find the -intercept, set = 0 in eqution of circle. ( ) + 9 = 9 = [es to see from grph] To find the -intercept, set = 0 in eqution of circle. + ( + ) = 9 ( + ) = 5 [sutrcting ] EVISION = ± 5 [tking squre root nd sutrcting ] = 0.7 or 5. ( dp)
9 Achievement Stndrd 909 (Clculus.5) Curves cn e sketched directl from the prmetric equtions if these equtions re given in stndrd form. Alterntivel, the prmetric equtions cn e solved simultneousl to eliminte the prmeter nd crete Crtesin coordintes. Emple Q. Sketch the curve whose prmetric equtions re = cos θ, = sin θ. A. These prmetric equtions re of the form = cos θ, = sin θ where =, ie the re the prmetric equtions of circle centre (0, 0), rdius. The grph is shown. Alterntivel ( eliminting θ) = cos θ cos θ = = sin θ sin θ = [rerrnging nd squring] [rerrnging nd squring] + = 1 [using Pthgors identit sin θ + cos θ = 1] + = [multipling ] which is circle, centre (0, 0), rdius = (s ove) The Ellipse The ellipse, centre the origin, with semi-mjor is, nd semi-minor is, hs Crtesin coordintes = e + = 1 = e nd prmetric equtions = cos θ + = 1 = sin θ semi-minor is F 1 semi-mjor is F ESA Pulictions (NZ) Ltd, Freephone 000-7 centre (0, 0) -intercepts (±, 0) -intercepts (0, ±) directrices = ± e foci (±e, 0) The ellipse + = 1 is the locus of point P (, ) moving such tht the sum of the distnces of P from two fied points F 1 nd F (the foci, singulr focus) is constnt (equl to ). The coordintes of the foci re (±e, 0) where e, numer etween 0 nd 1 is the eccentricit of the curve, nd ( ) = (1 e ). errnging gives the coordintes of the foci s ±, 0. The directrices (singulr directri) hve equtions = e nd = e. This ellipse m e trnslted, s discussed ove for the circle. Emples Q. 1. Sketch the grph of + = 1. 9 A. 1. Grph is n ellipse, centre (0, 0), where = = The grph is s shown. EVISION 9 = is the semi-mjor is, nd = is the semi-minor is. Q.. Sketch the grph of + + 1 + 1 = 0. A.. Convert eqution to stndrd form completing the squre: ( + ) + ( ) = 1 [comining terms in nd terms in nd isolting the constnt] ( + + 1) + ( + ) = 1 + 1 + [completing the squre]
Sketch grphs nd find equtions of conic sections 95 ( + 1) + ( ) = ( + 1) ( ) + = 1 [dividing to get 1 on HS] 1 which is n ellipse which hs een trnslted the vector 1 where = =, = 1 = 1 (ie semi-mjor is = nd semi-minor is = 1) The grph is s shown. 1 1 EVISION B emining the fetures of grph, its eqution cn e written down. In the ellipse shown =, = [semi-mjor nd semi-minor is lengths] Emple The ellipse is centred t (, ), so its eqution is ( ) ( ) + = 1 [using eqution of stndrd ellipse (trnslted)] ie ( ) ( ) + 9 = 1 9 Questions Sketching Conics the Circle nd the Ellipse 1. Sketch the grph of + + + 7 = 0. Lel n intercepts nd n smptotes.. Sketch the grph of ( ) + = 1. 9 Lel n intercepts nd n smptotes. QUESTIONS ESA Pulictions (NZ) Ltd, Freephone 000-7
9 Achievement Stndrd 909 (Clculus.5). Sketch the grph of + + + 1 = 0. Lel n intercepts nd smptotes.. Sketch the grph of the curve defined = cos t, = sin t. Lel fetures such s intercepts nd smptotes. QUESTIONS ESA Pulictions (NZ) Ltd, Freephone 000-7
Sketch grphs nd find equtions of conic sections 97 Achievement Stndrd 909 (Clculus.5) Sketch grphs of conic sections nd write equtions relted to conic sections.5 Sketching Conics the Circle nd the Ellipse (pge 95) 1... + ( + ) = 7 rerrnging eqution is ellipse + = 1 trnslted ( ) (A) 0 + ( + + 1) = 7 + 1 completing squre where semi-mjor is is + ( + ) = 9 stndrd form nd semi-minor is in eqution is circle centre (0, ) rdius (A). + + = 1 rerrnging ( + ) + ( 5) = 1 + 1 + 5 completing ( + ) + ( 5) = 5 the squre Circle centre (, 5) rdius 5 When = 0, + ( 5) = 5 ( 5) = 9 = 5 ± = or When = 0, ( + ) + ( 5) = 5 ( + ) = 0 = (A) ANSWES Prmetric equtions re of n ellipse with semimjor is nd semi-minor is, centre (0, 0). (A) compring with = cos θ, = sin θ the prmetric equtions of n ellipse ESA Pulictions (NZ) Ltd, Freephone 000-7