SOLVED EXAMPLES. be the foci of an ellipse with eccentricity e. For any point P on the ellipse, prove that. tan

Size: px

Start display at page:

Download "SOLVED EXAMPLES. be the foci of an ellipse with eccentricity e. For any point P on the ellipse, prove that. tan"

Roy McDaniel
5 years ago
Views:

1 LOCUS 58 SOLVED EXAMPLES Empl Lt F n F th foci of n llips with ccntricit. For n point P on th llips, prov tht tn PF F tn PF F Assum th llips to, n lt P th point (, sin ). P(, sin ) F F F = (-, 0) F = (, 0) Using th sin rul in PF, F w hv PF PF F F sin sin sin( ( )) PF PF F F sin sin sin( ) PF PF will lws qul sin sin sin( ) th lngth of th mjor is tn tn This is th sir rsult. Mthmtics / Ellips

2 LOCUS 59 Empl Lt th prpniculr istnc from th cntr of th llips th llips. If F n F r th two foci of th llips, prov tht to th tngnt rwn t point P on ( PF PF ) 4 Lt P th point (, sin ) whrs F n F r givn (,0). P F F B finition, th focl istnc of n point on n llips is tims th istnc of tht point from th corrsponing irctri. Thus, PF PF Now, th qution of th tngnt t P is ( ) 4 PF PF...() sin 0 Mthmtics / Ellips

3 LOCUS 60 Th istnc of (0,0) from this tngnt is. Thus, sin sin () From () n (), w s tht th qulit stt in th qustion os in hol. Empl Fin th rius of th lrgst circl with cntr (, 0) tht cn inscri insi th llips 6 4 Th following igrm shows th lrgst such circl. Osrv it crfull :. R O C(,0) R Not tht th lrgst possil circl ling compltl insi th llips must touch it, s, t th points R n R, s shown. At ths points, it will possil to rw common tngnts to th circl n th llips. Lt point R (4, sin ). Th qution of th tngnt t R is Mthmtics / Ellips sin 4 sin 4

4 LOCUS 6 If CR is prpniculr to this tngnt ( which must hppn if this tngnt is to common to oth th llips n th circl), w hv sin 0 4 sin Thus, R is Empl 4 4 4,. Th lrgst possil rius is thrfor r CR m A tngnt is rwn to th llips E : which cuts th llips E : t th points A n B. c Tngnts to this scon llips t A n B intrsct t right ngls. Prov tht c Lt th point of intrsction of th two tngnts P( h, k). A P( h,k) B E E Not tht sinc AB is th chor of contct for th tngnts rwn from P to E, w hv th qution of AB s Mthmtics / Ellips T ( h, k) 0 h k c h c k k

5 LOCUS 6 If AB is to touch th innr llips E, th conition of tngnc must stisfi : k h c k c h c k...() Sinc PA n PB intrsct t right ngls, P must li on th irctor circl of th llips E. Thus, h k c...() () n () cn consir sstm of qutions in th vrils h n k : ( ) h ( c ) k c h + k = c + If ths rltions r to hol for vril h n k, th must in fct inticl. Thus, ths vrils cn now sil limint to otin : 4 4 c ; c c c c c ; c c c c Empl 5 Lt ABC n quiltrl tringl inscri in th circl. Prpniculrs from A, B, C to th mjor is of th llips (whr > ) mt th llips rspctivl t P, Q, R so tht P, Q n R li on th sm si of th mjor is of A, B n C rspctivl. Prov tht th normls to th llips t P, Q n R r concurrnt. Th following figur grphicll portrs th sitution scri in th qustion : A B Q O R P W n to show tht th normls t P, Q n R r concurrnt C Mthmtics / Ellips

6 LOCUS 6 All th informtion givn out ABC ing quiltrl n ll cn ruc to this singl pic of significnt informtion : th polr ngls of A, B n C, n hnc, th ccntric ngls of P, Q n R, will vnl spc t, virtu of ABC ing quiltrl. Thus, w cn ssum th ccntric ngls of P, Q n R to,,. Now, th qution of norml to t ccntric ngl is givn sc c. ( )sin sin. Thus, th normls t P, Q n R r rspctivl givn N : sin P ( )sin 4 NQ : sin sin ( ) 4 NR : sin sin Lt us vlut, th trminnt of th cofficints of ths thr qutions : sin sin ( ) 4 sin sin 4 sin sin Using th row oprtion R R, R R th first row rucs to zro, which mns tht Thus, th normls t P, Q n R must concurrnt. 0 Mthmtics / Ellips

7 LOCUS 64 Empl 6 An llips slis twn two lins t right ngls to on nothr. Show tht th locus of its cntr is circl. It woul sist to ssum th two lins to th coorint is, n n llips of fi imnsions sliing twn ths two lins s shown low : S( h,k) Q O P Th llips hs fi imnsions, s, Mjor is = Minor is = Assum th cntr S to ( h, k) From th viw-point of th llips, sinc th tngnts to it t P n Q intrsct t right ngls t O, th point O must li on th irctor circl of th llips. Sinc th rius of th irctor circl is, w must hv OS OS h k This must th locus of th cntr S! It cn writtn in form s It is vint tht this is circl cntr t th origin n of rius. Empl 7 Fin th locus of th point P such tht tngnts rwn from it to th llips in concclic points. mt th coorints s Mthmtics / Ellips

8 LOCUS 65 P( h, k) A Th tngnts from P( h, k) to th llips mt th coorint s in A, B, C n D (not shown) which r concclic. B C Th pir of tngnts PA n PC hs th joint qution Th coorint s hs th joint qution J : T ( h, k) S(, ) S( h, k) J h k h k : 0 J : 0 W cn trt J n J s two curvs, which intrsct in four iffrnt points A, B, C, D. An scon gr curv through ths four points cn writtn in trms of prmtr s W now simpl fin tht concclic. This rprsnts circl if J J 0 for which this rprsnts circl, sinc A, B, C, D r givn to h k h k 0 k h () itslf givs th locus of P(h, k) s h k...() hk...() Mthmtics / Ellips

9 LOCUS 66 Empl 8 Through n ritrr fi point P ( ) on th llips, chors t right ngls r rwn, such tht th lin joining th trmitis of ths chors mts th norml through P t th point Q. Prov tht Q is fi for ll such chors. P( ) A Q B W cn ssum th ccntric ngls of A n B s n. Th norml t P hs th qution : Th chor AB hs th qution Also, sinc PA PB, w hv PQ : sc c...() AB : sin...() sin sin slop of chor PA slop of chor PB sin sin 0 Using trignomtric formul, this prssion cn rrrng to Mthmtics / Ellips...()

10 LOCUS 67 From () n (), w hv AB : sin...(4) Th point Q cn now otin s th intrsction of th lins rprsnt () n (4). Lt us writ thm s sstm n solv for Q using th Crmr s rul: PQ : ( sc ) ( c ) ( ) 0 ( ) AB : ( ) ( sin ) ( ) 0 whr hs n sustitut for convninc. W now hv ( ) ( ) c ( ) sin ( ) ( ) sc ( ) sc sin c ( ) ( ) sin sin ( ) sin ( sc sin c ) ( )...(5) n ( ) ( ) ( ) ( sc sin c ) ( )sin...(6) Thus, th point Q, whos n coorints r givn (5) n (6) rspctivl, cn sn to inpnnt of or. pns onl on th ccntric ngl of P. Q is th thrfor fi for such pirs of chors PA n PB n Mthmtics / Ellips

11 LOCUS 68 Empl 9 Consir thr points on th llips, P( ), Q( ) is this r mimum? Th thr points hv th coorints P (, sin ); Q (, sin ); R (, sin ) Th r of this tringl, th trminnt formul, is n R ( ). Wht is th r of PQR? Whn sin sin sin...() (sin sin ) sin ( ) sin sin sin( ) sin( ) sin( ) sin( ) sin sin sin sin sin This is th r of th tringl PQR. To fin its mimum vlu, w us rthr inirct rout. Suppos w h to clcult th r tringl inscri in th circl with th sm polr ngls s P, Q, R. Th onl iffrnc twn n will tht in th trminnt prssion for in (), w will hv ll inst of in th trms of th scon column. This mns tht n will lws in constnt rtio : of Thus, th mimum for of chiv! will chiv in th sm configurtion s th on in which th mimum Mthmtics / Ellips

12 LOCUS 69 Sinc th r of tringl inscri in circl hs th mimum vlu whn tht tringl is quiltrl (this shoul intuitivl ovious ut cn lso sil prov), n hnc will mimum whn Thus, th thr ccntric ngls must qull spc prt t. Empl 0 Prov tht th circl on n focl istnc s imtr touchs th uilir circl of th llips. Lt P ( ) n ritrr point on th llips n lt F on of its foci. P F C O Auilir circl Th rius of th uilir is. Th circl on PF s imtr will touch th uilir circl (intrnll) if : OC (rius of this circl) C, ing th mi-point of PF, hs th coorints C sin, Mthmtics / Ellips

13 LOCUS 70 Thus, OC sin sin ( ) sin Also, th rius of th innr circl is ( ) CF sin ( )sin ( ) This givs which provs th stt ssrtion. OC CF Mthmtics / Ellips

14 LOCUS 7 A S S I G N M E N T [ LEVEL - I ] If th normls t th points, n on th llips sc c sc c 0 sc c r concurrnt, prov tht Th tngnt n norml t point P on th llips mt th minor is t A n B rspctivl. Prov tht th circl with AB s imtr psss through P s wll s th two foci of th llips Lt AB focl chor of th llips th locus of P.. Th tngnt t A n th norml t B intrsct in P. Fin A lin intrscts th llips t A n B n th prol 4 ( ) t C n D. AB sutns right ngl t th cntr of th llips. Tngnts to th prol t C n D intrsct in E. Prov tht th locus of E is 4 4 ( ) Prov tht th common chors of n llips n circl r qull inclin to th s of th llips. Prov tht th minimum lngth of th intrcpt m th s on th tngnts to th llips is qul to +. If is th lngth of th prpniculr from th focus F of llips, prov tht FP upon tngnt t n point P on th With givn point n lin s focus n irctri, sris of llipss r scri. Prov tht th locus of th trmitis of thir minor is is prol. Th tngnt t n point P on th llips mts its uilir circl in A n B. AB sutns right ngl t th cntr of th llips. Lt th ccntricit of th llips n th ccntric ngl of P. Show tht sin Show tht th common tngnt to th llipss sutns right ngl t th origin. c n c 0 Mthmtics / Ellips

15 LOCUS 7 [ LEVEL - II ] If vril point P on n llips is join to th two foci F n F, show tht th incntr of PF F lis on nothr llips. Two tngnts to givn llips intrsct t right ngls. Prov tht th sum of th squrs of th chors which th uilir circl intrcpts on ths tngnts, is constnt. From n point P on th llips E : ( ), tngnts r rwn to th llips E :, which touch E t A n B. Prov tht th orthocntr of PAB lis on E. Normls to th llips t th points (, ), i,,, r concurrnt. Show tht i i 0 A point P movs so tht circl with PQ (whr Q (, 0)) s imtr touchs th circl 4 intrnll. Prov tht th locus of P is n llips. Normls t th points,, n on th llips 4 ( 4) (sc sc sc sc 4 ) 4 r concurrnt. Prov tht Tngnts r rwn from n point on mi-point of th chor of contct. 6 9 to th circl. Fin th locus of th A prlllogrm circumscris th llips lins c. Fin th locus of th othr two vrtics of th prlllogrm. n two of its opposit vrtics li on th stright Th s of vril tringl is of fi lngth n th sum of its si, s, is lso fi. Prov tht th locus of th incntr of th tringl is n llips. Lt P ( ) point on th llips. A prol is rwn hving its focus t P n pssing through th foci of th givn llips. Show tht two such prols cn rwn. Prov lso tht thir irctrics will inclin t n ngl. Mthmtics / Ellips

16 LOCUS 7 A S S I G N M E N T ( ANSWERS ) [ LEVEL - I ] ( ) [ LEVEL - II ] 6 9 ( ) c c Mthmtics / Ellips

17 LOCUS 74 ANSWERS TRY YOURSELF - I 9 5, TRY YOURSELF - II sc TRY YOURSELF - III 4 4 c TRY YOURSELF - IV, Mthmtics / Ellips

CONIC SECTIONS. MODULE-IV Co-ordinate Geometry OBJECTIVES. Conic Sections

CONIC SECTIONS. MODULE-IV Co-ordinate Geometry OBJECTIVES. Conic Sections Conic Sctions 16 MODULE-IV Co-ordint CONIC SECTIONS Whil cutting crrot ou might hv noticd diffrnt shps shown th dgs of th cut. Anlticll ou m cut it in thr diffrnt ws, nml (i) (ii) (iii) Cut is prlll to