Notes for Geometry Conic Sections


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1 Nots for Gomtry Conic Sctions Th nots is takn from Gomtry, by David A. Brannan, Matthw F. Espln and Jrmy J. Gray, 2nd dition 1 Conic Sctions A conic sction is dfind as th curv of intrsction of a doubl con with a plan. Figur: Exampls: 1. nondgnrat conic sctions: parabolas, llipss or hyprbolas; 2. dgnrat conic sctions: th singl point, singl lin and pair of lins. 1.1 FocusDirctrix Dfinition of th NonDgnrat Conics Th 3 nondgnrat conics can b dfind as th st of points P in th plan satisfying: Th distanc of P from a fixd point F (calld th focus of th conic) is a constant multipl (calld its ccntricity, ) of th distanc of P from a fixd lin d (calld its dirctrix). Eccntricity: A nondgnrat conic is 1. an llips if 0 < 1, 2. a parabola if = 1, 3. a hyprbola if > 1. 1
2 Parabola ( = 1): A parabola is dfind to b th st of points P in th plan whos distanc from a fixd point F is qual to thir distanc from a fixd lin d. W now driv a parabola in standard form: Lt F (a, 0) b th focus and d : x = a b th dirctrix. Lt P (x, y) b an arbitrary point on th parabola and lt M( a, y) b th foot of th prpndicular from P to th dirctrix. Figur: Sinc F P = P M, by th dfinition of th parabola, this follows that F P 2 = P M 2 (x a) 2 + y 2 = (x + a) 2 y 2 = 4ax. Th point (at 2, 2at), t R lis on th parabola. (2at) 2 = 4a at 2. Convrsly, w can writ th coordinats of ach point on th parabola in th form (at 2, 2at). For if w choos t = y y2, thn y = 2at and x = = (2at)2 = at 2. It follows 2a 4a 4a that thr is a ontoon corrspondnc btwn th ral numbrs t and th points of th parabola. Parabola in standard form A parabola in standard form has quation y 2 = 4ax, whr a > 0. It has focus (a, 0) and dirctrix x = a; and it can b dscribd by th paramtric quations: x = at 2, y = 2at (t R). W call th xaxis th axis of th parabola in standard form, sinc th parabola is symmtric with rspct to this lin. W call th origin th vrtx of a parabola in standard form, sinc it is th point of th intrsction of th axis of th parabola with th parabola. A parabola has no cntr. 2
3 Exampl 1.1. Writ down th focus, vrtx, axis and dirctrix of th parabola E with quation y 2 = 2x. Solution. Focus: F = ( 1 2, 0), Axis: xaxis, Vrtx: (0, 0), Dirctrix: x = 1 2. Ellips (0 < 1): W dfin an llips with ccntricity zro to b a circl. W dfin an llips with ccntricity (whr 0 < < 1) to b th st of points P in th plan whos distanc from a fixd point F is tims thir distanc from a fixd lin. W now driv an llips in standard form: Lt F (a, 0), a > 0 b th focus and d : x = a. Lt P (x, y) b an arbitrary point on th parabola and lt M( a, y) b th foot of th prpndicular from P to th dirctrix. Figur: Sinc F P = P M, by th dfinition of th llips, this follows that F P 2 = 2 P M 2 (x a) 2 + y 2 = 2 (x a )2 x2 a 2 + y 2 a 2 (1 2 ) = 1. Lt b = a 1 2. Thn a + y2 2 b = 1. 2 Th quation is symmtrical in x and y. Th llips also has a scond focus F ( a, 0) and a scond dirctrix d : x = a. W call th sgmnt joining th points (±a, 0) th major axis of th llips and th sgmnt joining th points (0, ±b) th minor axis of th llips. b < a, th minor axis is shortr than th major axis. Th origin is th cntr of this llips. Not that (a cos t, b sin t) lis on th llips. (a cos t)2 (b sin t)2 + = cos 2 t + sin 2 t = 1. a 2 b 2 3
4 W can chck that x = a cos t, y = b sin t, t ( π, π] givs a paramtric rprsntation of th llips. Ellips in standard form An llips in standard form has quation a 2 + y2 b 2 = 1, whr a b > 0, b2 = a 2 (1 2 ), 0 < 1. It can b dscribd by th paramtric quations x = a cos t, y = b sin t, t ( π, π]. If > 0, it has foci (±a, 0) and dirctrix x = ± a. Exrcis 1.2. (x + a)2 + y 2 + (x a) 2 + y 2 = 2a. Hyprbola ( > 1): A hyprbola is th st of points P in th plan whos distanc from a fixd point F is tims thir distanc from a fixd lin d, whr > 1. W now driv a hyprbola in standard form: Lt F (a, 0), a > 0 b th focus and d : x = a. Lt P (x, y) b an arbitrary point on th parabola and lt M( a, y) b th foot of th prpndicular from P to th dirctrix. Figur: Sinc F P = P M, by th dfinition of th hyprbola, this follows that F P 2 = 2 P M 2 (x a) 2 + y 2 = 2 (x a )2 x2 a 2 y 2 a 2 ( 2 1) = 1. Lt b = a 2 1. Thn a 2 y2 b 2 = 1. 4
5 Th quation is symmtrical in x and y. Th llips also has a scond focus F ( a, 0) and a scond dirctrix d : x = a. W call th sgmnt joining th points (±a, 0) th major (transvrs) axis of th hyprbola and th sgmnt joining th points (0, ±b) th minor (conjugat) axis of th hyprbola. Th origin is th cntr of this hyprbola. W can chck that x = sc t, y = b tan t, t ( π, π) ( π, 3π ) givs a paramtric rprsntation of th hyprbola. Th hyprbola consist of 2 branchs. Whn x ±, th branchs gt closr and closr to th lins y = ± b x th asymptots of th a hyprbola. Hyprbola in standard form A hyprbola in standard form has quation a 2 y2 b 2 = 1, whr a b > 0, b2 = a 2 ( 2 1), > 1. it has foci (±a, 0) and dirctrix x = ± a. It can b dscribd by th paramtric quations x = a sc t, y = b tan t, t ( π 2, π 2 ) (π 2, 3π 2 ). Exampl 1.3. Dtrmin th foci F and F of th hyprbola E with quation 2y 2 = 1. Solution. ( ± ) 3/2, 0. Exrcis 1.4. (x + a)2 + y 2 (x a) 2 + y 2 = ±2a. 5
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