ELLIPSE. Standard equation of an ellipse referred to its principal axes along the co-ordinate axes is. ( a,0) A'

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1 J-Mthemtics LLIPS. STANDARD QUATION & DFINITION : Stndrd eqution of n ellipse referred to its principl es long the co-ordinte es is > & = ( e ) = e. Y + =. where where e = eccentricit (0 < e < ). FOCI : S (e, 0) & S' ( e, 0 ). ( ) qution of directrices : = e & ( ) Vertices :. e X' 0 M' e Z' Directri (,0) A' L B(0,) L S' ( e,0) C S A(,0) Z L ' L' B'(0, ) A' (, 0) & A (, 0). Y' ( c ) Mjor is : The line segment A' A in which the foci S' & S lie is of length & is clled the mjor is ( > ) of the ellipse. Point of intersection of mjor is with directri is clled the foot of the directri (z), 0 e. ( d ) Minor Ais : The -is intersects the ellipse in the points B (0,- ) & B (0, ). The line segment B B of length ( < ) is clled the Minor Ais of the ellipse. ( e) Principl Aes : The mjor & minor is together re clled Principl Aes of the ellipse. ( f ) Centre : The point which isects ever chord of the conic drwn through it is clled the centre of the conic. C (0,0) the origin is the centre of the ellipse. ( g ) Dimeter : A chord of the conic which psses through the centre is clled dimeter of the conic. (h ) Focl Chord : A chord which psses through focus is clled focl chord. ( i ) Doule Ordinte : A chord perpendiculr to the mjor is is clled doule ordinte. ( j ) Ltus Rectum : The focl chord perpendiculr to the mjor is is clled the ltus rectum. (i) Length of ltus rectum (LL') = (ii) (iii) qution of ltus rectum : = ± e. nds of the ltus rectum re ( m in or is ) m jor is ( e ) L e,, L ' e,, L e, ( k ) Focl rdii : SP = e & S'P = + e SP + S 'P = = Mjor is. ( l ) ccentricit : Note : (i) (ii) e nd Directri M = e L ' e,. The sum of the focl distnces of n point on the ellipse is equl to the mjor Ais. Hence distnce of focus from the etremit of minor is is equl to semi mjor is. i.e BS = CA. If the eqution of the ellipse is given s tht >. & nothing is mentioned, then the rule is to ssume X NOD6\\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#09\ng\0 LLIPS.p65

2 J-Mthemtics Illustrtion : If LR of n ellipse is hlf of its minor is, then its eccentricit is - Solution : Illustrtion : (A) (B) As given = 4 = 4 ( e ) = e = /4 e = / Ans. (C) Find the eqution of the ellipse whose foci re (, ), (, ) nd whose semi minor is is of length 5. Solution : Here S is (, ) & S' is (, ) nd = 5 SS' = 4 = e e = ut = ( e ) 5 = 4 =. Hence the eqution to mjor is is = Centre of ellipse is midpoint of SS' i.e. (0, ) ( ) ( ) qution to ellipse is or Ans. 9 5 Illustrtion : Find the eqution of the ellipse hving centre t (, ), one focus t (6, ) nd pssing through the point (4, 6). Solution : With centre t (, ), the eqution of the ellipse is Do ourself - : (i) point (4, 6) (C) (D). It psses through the (i) Distnce etween the focus nd the centre = (6 ) = 5 = e = e = 5... (ii) Solving for nd from the equtions (i) nd (ii), we get = 45 nd = 0. Hence the eqution of the ellipse is If LR of n ellipse Ans. 45 0, ( < ) is hlf of its mjor is, then find its eccentricit. (ii) Find the eqution of the ellipse whose foci re (4, 6) & (6, 6) nd whose semi-minor is is 4. (iii) Find the eccentricit, foci nd the length of the ltus-rectum of the ellipse = 0. NOD6\\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#09\ng\0 LLIPS.p65. ANOTHR FORM OF LLIPS : ( ) AA' = Minor is = ( ) BB' = Mjor is = ( c ) = ( e ) ( d ) Ltus rectum LL' = L L ' ( e) nds of the ltus rectum re :, ( <), eqution = ± e L, e, L ', e, L, e, L ', e ( f ) qution of directri = ± /e ( g ) ccentricit : e X' ( ), e Directri (,0) A' L' Directri (0,e) Y Z = /e B(0,) S C (0,0) (0, e) S' L ' Z' Y' B'(0, ) L L = /e ( ), e A(,0) X'

3 J-Mthemtics Illustrtion 4 : Solution : The eqution of the ellipse with respect to coordinte es whose minor is is equl to the distnce etween its foci nd whose LR = 0, will e- (A) + = 00 (B) + = 00 (C) + = 80 (D) none of these When > As given = e = e... (i) Also 0 = 5... (ii) Now since = e = [From (i)] =... (iii) (ii), (iii) = 00, = 50 Hence eqution of the ellipse will e + = Similrl when < then required ellipse is + = 00 Ans. (A, B) Do ourself - : (i) The foci of n ellipse re (0, ±) nd its eccentricit is. Find its eqution (ii) Find the centre, the length of the es, eccentricit nd the foci of ellipse = 0 (iii) The eqution 8 t t 4, will represent n ellipse if (A) t (, 5) (B) t (, 8) (C) t (4, 8) {6} (D) t (4, 0) {6}. GNR AL QUATION OF AN LLIPS Let (, ) e the focus S, nd l + m + n = 0 is the eqution of directri. Let P(, ) e n point on the ellipse. Then definition. ( l m n) SP = e PM (e is the eccentricit) ( ) + ( ) = e ( l m ) (l + m ) {( ) + ( ) } = e {l + m + n} S(,) P'(,) is M l + m + n = 0 4. POSITION OF A POINT W.R.T. AN LLIPS : The point P(, ) lies outside, inside or on the ellipse ccording s ; 5. AUXILLIARY CIRCL/CCNTRIC ANGL : A circle descried on mjor is s dimeter is clled the uilir circle. Let Q e point on the uilir circle + = such tht QP produced is perpendiculr to the -is then P & Q re clled s the CORRSPONDING POINTS on the ellipse & the uilir circle respectivel. is clled the CCNTRIC ANGL of the point P on the ellipse (0 < ). Note tht l(pn) Semi minor is = l(qn) Semi mjor is > < or = 0. Y Q P A' S' O N S (,0) A(, 0) Hence If from ech point of circle perpendiculrs re drwn upon fied dimeter then the locus of the points dividing these perpendiculrs in given rtio is n ellipse of which the given circle is the uilir circle. NOD6\\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#09\ng\0 LLIPS.p65

4 J-Mthemtics 6. PAR A MTRIC RPRSNTATION : The equtions = cos & = sin together represent the ellipse where is prmeter (eccentric ngle). Note tht if P( ) ( cos, sin ) is on the ellipse then ; Q( ) ( cos, sin) is on the uilir circle. 7. LIN AND AN LLIPS : The line = m + c meets the ellipse c is < = or > m +. in two points rel, coincident or imginr ccording s Hence = m + c is tngent to the ellipse if c = m +. The eqution to the chord of the ellipse joining two points with eccentric ngles & is given cos s in cos. Illustrtion 5 : For wht vlue of does the line = + touches the ellipse = 44. Solution : qution of ellipse is = 44 or 6 9 Compring this with then we get = 6 nd = 9 nd compring the line = + with = m + c m = nd c = If the line = + touches the ellipse = 44, then c = m + = = 5 = ± 5 A ns. NOD6\\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#09\ng\0 LLIPS.p65 Illustrtion 6 : If, re eccentric ngles of end points of focl chord of the ellipse tn /. tn / is equl to - (A) e (B) e (C) e (D) e e e e e Solution : qution of line joining points nd is cos sin cos If it is focl chord, then it psses through focus (e, 0), so e cos cos cos cos cos e e e cos cos cos sin / sin / e tn cos / cos / e e tn e, then e using ( e, 0), we get tn tn Ans. (A,C) e

5 J-Mthemtics Do ourself - : (i) Find the position of the point (4, ) reltive to the ellipse + 9 =. (ii) A tngent to the ellipse, ( > ) hving slope intersects the is of & in point A & B (iii) respectivel. If O is the origin then find the re of tringle OAB. Find the condition for the line cos + sin = P to e tngent to the ellipse. 8. TANGNT TO TH LLIPS + = ( ) Point form : qution of tngent to the given ellipse t its point (, ) is : Note : For generl ellipse replce ( ), ( ), ( + ), ( + ), ( + ) & c (c). ( ) Slope form : qution of tngent to the given ellipse whose slope is 'm', is = m ± m Point of contct re m, m m Note tht there re two tngents to the ellipse hving the sme m, i.e. there re two tngents prllel to n given direction. ( c ) Prmetric form : qution of tngent to the given ellipse t its point ( cos, sin ), is cos s in Note : (i) The eccentric ngles of point of contct of two prllel tngents differ. (ii) Point of intersection of the tngents t the point & is cos cos s in, cos Illustrtion 7 : Find the equtions of the tngents to the ellipse + 4 = which re perpendiculr to the line + = 4. Solution : Let m e the slope of the tngent, since the tngent is perpendiculr to the line + = 4. m = m = Since + 4 = or Compring this with = 4 nd = 4 So the eqution of the tngent re 4 4 = ± or ± 4 = 0. A ns. 4 NOD6\\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#09\ng\0 LLIPS.p65

6 J-Mthemtics Illustrtion 8 : Solution : The tngent t point P on n ellipse intersects the mjor is in T nd N is the foot of the perpendiculr from P to the sme is. Show tht the circle drwn on NT s dimeter intersects the uilir circle orthogonll. Let the eqution of the ellipse e. Let P(cos, sin) e point on the ellipse. The eqution of the tngent t P is cos sin. It meets the mjor is t T ( sec, 0). The coordintes of N re ( cos, 0). The eqution of the circle with NT s its dimeter is ( sec)( cos) + = 0. + (sec + cos) + = 0 It cuts the uilir circle + = 0 orthogonll if g. 0 + f. 0 = = 0, which is true. A ns. Do ourself - 4 : (i) Find the eqution of the tngents to the ellipse = 44 which re prllel to the line + + k = 0. (ii) Find the eqution of the tngent to the ellipse = 00 t the point (, ). 9. NORMAL TO TH LLIPS + = ( ) Point form : qution of the norml to the given ellipse t (, ) is : = = e. ( ) Slope form : qution of norml to the given ellipse whose slope is m is = m ( ) m. m ( c ) Prmetric form : qution of the norml to the given ellipse t the point (cos, sin) is sec cosec = ( ). Illustrtion 9 : Find the condition tht the line + m = n m e norml to the ellipse. Solution : qution of norml to the given ellipse t ( cos, sin ) is cos sin...(i) NOD6\\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#09\ng\0 LLIPS.p65 If the line + m = n is lso norml to the ellipse then there must e vlue of for which line (i) nd line + m = n re identicl. For tht vlue of we hve m n cos sin ( ) n nd sin = m( ) Squring nd dding (iii) nd (iv), we get n cos ( ) or n ( ) m Illustrtion 0 : If the norml t n end of ltus-rectum of n ellipse of the minor is, show tht the eccentricit of the ellipse is given... (iii)... (iv) which is the required condition. psses through one etremit e 5 5

7 J-Mthemtics Solution : The co-ordintes of n end of the ltus-rectum re (e, /). The eqution of norml t P(e, /) is () or e / e It psses through one etremit of the minor is whose co-ordintes re (0, ) 0 + = ( ) = ( ). ( e ) = ( e ) e = e 4 e 4 + e = 0 (e ) + e = 0 4 e 6 (,0)A' (0,) B B'(0, ) P(e, /) S A(,0) 5 e (tking positive sign) A ns. Illustrtion : P nd Q re corresponding points on the ellipse nd the uilir circles respectivel. The norml t P to the ellipse meets CQ in R, where C is the centre of the ellipse. Prove tht CR = + Solution : Let P (cos, sin) Q Q (cos, sin) qution of norml t P is P (sec) (cosec) =... (i) C eqution of CQ is = tn.... (ii) Solving eqution (i) & (ii), we get ( ) = ( )cos = ( + ) cos, & = ( + ) sin R (( + )cos, ( + )sin CR = + A ns. Do ourself - 5 : (i) Find the eqution of the norml to the ellipse = 88 t the point (4, ) (ii) (iii) Let P e vrile point on the ellipse PF F, then find mimum vlue of A. If the norml t the point P() to the ellipse cos with foci F nd F. If A is the re of the tringle intersects it gin t the point Q(), then find (iv) Show tht for ll rel vlues of 't' the line t + t = touches fied ellipse. Find the eccentricit of the ellipse. 0. CHORD OF CONTACT : If PA nd PB e the tngents from point P(, ) to the ellipse. The eqution of the chord of contct AB is or T = 0 (t, ). Illustrtion : If tngents to the prol = 4 intersect the ellipse Solution : of point of intersection of tngents t A nd B. t A nd B, the find the locus Let P (h, k) e the point of intersection of tngents t A & B h k eqution of chord of contct AB is... (i) which touches the prol. NOD6\\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#09\ng\0 LLIPS.p65

8 J-Mthemtics qution of tngent to prol = 4 is = m + m m = m eqution (i) & (ii) s must e sme m m h k m = & m h k k 4 h k locus of P is =. k... (ii) P(h, k) A ns. Do ourself - 6 : (i) (ii) (iii) Find the eqution of chord of contct to the ellipse t the point (, ). 6 9 If the chord of contct of tngents from two points (, ) nd (, ) to the ellipse ngles, then find. re t right If line = intersects ellipse t points A & B, then find co-ordintes of point of 8 4 intersection of tngents t points A & B.. PAIR OF TANGNTS : Y If P(, ) e n point lies outside the ellipse, nd pir of tngents PA, PB cn e drwn to it from P. Then the eqution of pir of tngents of PA nd PB is SS = T where S, T i.e. = P (, ) X' A B C Y' X NOD6\\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#09\ng\0 LLIPS.p65. DIRCTOR CIRCL : Locus of the point of intersection of the tngents which meet t right ngles is clled the Director Circle. The eqution to this locus is + = + i.e. circle whose centre is the centre of the ellipse & whose rdius is the length of the line joining the ends of the mjor & minor is. Illustrtion : A tngent to the ellipse Solution : + 4 = 4 meets the ellipse + = 6 t P nd Q. Prove tht the tngents t P nd Q of the ellipse + = 6 re t right ngles. Given ellipse re... (i) 4 nd,... (ii) 6 n tngent to (i) is cos sin... (iii) It cuts (ii) t P nd Q, nd suppose tngent t P nd Q meet t (h, k) Then eqution of chord of contct of (h, k) with respect to ellipse (ii) is h k... (iv) 6 7

9 J-Mthemtics compring (iii) nd (iv), we get cos sin h k cos nd sin h k 9 h / k / locus of the point (h, k) is + = 9 + = 6 + = + i.e. director circle of second ellipse. Hence the tngents re t right ngles.. QUATION OF CHORD WITH MID POINT (, ) : The eqution of the chord of the ellipse where T,, whose mid-point e (, ) is T = S S, i.e. Illustrtion 4 : Find the locus of the mid-point of focl chords of the ellipse Solution : Let P (h, k) e the mid-point. h k h k eqution of chord whose mid-point is given since it is focl chord, It psses through focus, either (e, 0) or ( e, 0) If it psses through (e, 0) e locus is If it psses through ( e, 0) S P(h, k) locus is e A ns. Do ourself - 7 : (i) Find the eqution of chord of the ellipse whose mid point e (, ) IMPORTANT POINTS : Referring to n ellipse ( ) If P e n point on the ellipse with S & S' s its foci then ( S P ) ( S ' P ). ( ) The tngent & norml t point P on the ellipse isect the eternl & internl ngles etween the focl distnces of P. This refers to the well known reflection propert of the ellipse which sttes tht rs from one focus re reflected through other focus & vice vers. 8 X' Q B Y Tngent P A' S' C N Reflected r Y' B' S Norml Light r ( c ) The product of the length s of the perpendiculr segments from the foci on n tngent to the ellipse is nd the feet of these perpendiculrs lie on its uilir circle nd the tngents t these feet to the uilir circle meet on the ordinte of P nd tht the locus of their point of intersection is similr ellipse s tht of the originl one. ( d ) The portion of the tngent to n ellipse etween the point of contct & the directri sutends right ngle t the corresponding focus. X NOD6\\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#09\ng\0 LLIPS.p65

10 (e) J-Mthemtics If the norml t n point P on the ellipse with centre C meet the mjor & minor es in G & g respectivel, & if CF e perpendiculr upon this norml, then (i) PF. PG = (ii) PF. Pg = (iii) PG. Pg = SP. S P (iv) CG. CT = CS ( v ) locus of the mid point of Gg is nother ellipse hving the sme eccentricit s tht of the originl ellipse. [where S nd S' re the focii of the ellipse nd T is the point where tngent t P meet the mjor is] ( f ) Atmost four normls & two tngents cn e drwn from n point to n ellipse. ( g ) The circle on n focl distnce s dimeter touches the uilir circle. ( h ) Perpendiculrs from the centre upon ll chords which join the ends of n perpendiculr dimeters of the ellipse re of constnt length. (i) If the tngent t the point P of stndrd ellipse meets the es in T nd t nd CY is the perpendiculr on it from the centre then, (i) Tt. PY = nd (ii) lest vlue of Tt is +. Do ourself - 8 : (i) (ii) A mn running round rcecourse note tht the sum of the distnce of two flg-posts from him is lws 0 meters nd distnce etween the flg-posts is 6 meters. Find the re of the pth e encloses in squre meters If chord of contct of the tngent drwn from the point () to the ellipse + = k, then find the locus of the point (). Miscellneous Illustrtion : touches the circle Illustrtion 5 : A point moves so tht the sum of the squres of its distnces from two intersecting stright lines is constnt. Prove tht its locus is n ellipse. Solution : Let two intersecting lines OA nd OB, intersect t origin O nd let oth lines OA nd OB mkes equl ngles with is. i.e., XOA = XOB =. qutions of stright lines OA nd OB re N A NOD6\\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#09\ng\0 LLIPS.p65 = tn nd = tn or sin cos = 0... (i) nd sin + cos = 0... (ii) Let P( ) is the point whose locus is to e determine. According to the emple (PM) + (PN) = ( sin + cos) + ( sin cos) = sin + cos = or sin + cos = Hence required locus is (s) cosec sec ( cosec ) ( sec ) Illustrtion 6 : Find the condition on '' nd '' for which two distinct chords of the ellipse through (, ) re isected the line + =. O M P( ) ( cosec ) ( sec ) B A ns. pssing 9

11 J-Mthemtics Solution : Let (t, t) e point on the line + =. Then eqution of chord whose mid point (t, t) is t ( t) t ( t)... (i) (, ) lies on (i) then t ( t) t ( t) t ( + ) ( + )t + = 0 Since t is rel B 4AC 0 ( + ) 4( + ) , which is the required condition. Illustrtion 7 : An tngent to n ellipse is cut the tngents t the ends of the mjor is in T nd T '. Prove tht circle on TT' s dimeter psses through foci. Solution : Let ellipse e T' P nd let P(cos, sin) e n point on this ellipse qution of tngent t P(cos, sin) is cos sin...(i) The two tngents drwn t the ends of the mjor is re = nd = A' O T A Solving (i) nd = we get ( cos ) T,, tn sin nd solving (i) nd = we get ( cos ) T ',, cot sin qution of circle on TT' s dimeter is ( )( + ) + ( tn(/))( cot (/)) = 0 or + (tn(/) + cot(/)) + = 0... (ii) Now put = ± e nd = 0 in LHS of (ii), we get e = + = 0 = RHS Hence foci lie on this circle Illustrtion 8 : A vrile point P on n ellipse of eccentricit e, is joined to its foci S, S'. Prove tht the locus of the Solution : incentre of the tringle PSS' is n ellipse whose eccentricit is Let the given ellipse e Let the co-ordintes of P re ( cos, sin) B hpothesis = ( e ) nd S(e, 0), S'( e, 0) nd SP = focl distnce of the point P = e cos S'P = + e cos Also SS' = e 40 e e. (+e cos ) P (cos sin ) I(, ) ( e cos ) S' e S ( e, 0) (e, 0) NOD6\\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#09\ng\0 LLIPS.p65

12 J-Mthemtics If (, ) e the incentre of the PSS' then (e) cos ( e cos )( e) ( e cos )e = e ( e cos ) ( e cos ) = e cos... (i) e( sin ) ( e cos ).0 ( e cos ).0 = e ( e cos ) ( e cos ) e sin (e )... (ii) liminting from equtions (i) nd (ii), we get which represents n ellipse. Let e e its eccentricit. e (e ) e ( e ) e (e ) e e e = e e e (e ) e e e e e ANSWRS FOR DO YOURSLF : (i) e (ii) ( 0) ( 6) (iii) 5 6 e ; foci (, ); LR : (i) 4 8 NOD6\\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#09\ng\0 LLIPS.p65 (ii) C (, ), length of mjor is = =, length of minor is = = ; e= ; ƒ, (iii) C : (i) On the ellipse (ii) ( ) 4 : (i) + ± 97 = 0 (ii) 7 = 50 (iii) P = cos + sin 5 : (i) 4 = 7 (ii) e (iii) (iv) 6 : (i) (ii) 6 7 : (i) = 5 8 : (i) 60 (ii) k (iii) (, ) 4

13 J-Mthemtics XRCIS - 0 CHCK YOUR GRASP SLCT TH CORRCT ALTRNATIV (ONLY ON CORRCT ANSWR). If distnce etween the directrices e thrice the distnce etween the foci, then eccentricit of ellipse is - (A) (B). If the eccentricit of n ellipse e 5/8 nd the distnce etween its foci e 0, then its ltus rectum is - (A) 9 4 (C) (D) 4 5 (B) (C) 5 (D) 7. The curve represented = (cost + sint), = 4(cost sint), is - (A) ellipse (B) prol (C) hperol (D) circle 4. If the distnce of point on the ellipse from the centre is, then the eccentric ngle is- 6 (A) / (B) /4 (C) /6 (D) / 5. An ellipse hving foci t (, ) nd ( 4, 4) nd pssing through the origin hs eccentricit equl to- (A) 7 (B) 7 (C) 5 7 (D) 5 6. A tngent hving slope of 4 to the ellipse intersects the mjor & minor es in points A & B 8 respectivel. If C is the centre of the ellipse then the re of the tringle ABC is : (A) sq. units (B) 4 sq. units (C) 6 sq. units (D) 48 sq. units 7. The eqution to the locus of the middle point of the portion of the tngent to the ellipse 6 + = included 9 etween the co-ordinte es is the curve- (A) = 4 (B) = 4 (C) + 4 = 4 (D) = 8. An ellipse is drwn with mjor nd minor es of lengths 0 nd 8 respectivel. Using one focus s centre, circle is drwn tht is tngent to the ellipse, with no prt of the circle eing outside the ellipse. The rdius of the circle is- (A) (B) (C) (D) 5 9. Which of the following is the common tngent to the ellipses 4 (A) = (B) = 4 4 (C) = 4 4 (D) = Angle etween the tngents drwn from point (4, 5) to the ellipse is (A) (B) 5 6 (C) 4 & (D)?. The point of intersection of the tngents t the point P on the ellipse + =, nd its corresponding point Q on the uilir circle meet on the line - (A) = /e (B) = 0 (C) = 0 (D) none NOD6\\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#09\ng\0 LLIPS.p65

14 NOD6\\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#09\ng\0 LLIPS.p65 (D) foci re ( 7, ) 4 J-Mthemtics. An ellipse is such tht the length of the ltus rectum is equl to the sum of the lengths of its semi principl es. Then - (A) llipse ecomes circle (C) llipse ecomes prol (B) llipse ecomes line segment etween the two foci (D) none of these. The eqution of the norml to the ellipse t the positive end of ltus rectum is - (A) + e + e = 0 (B) e e = 0 (C) e e = 0 (D) none of these 4. The eccentric ngle of the point where the line, 5 = 8 is norml to the ellipse (A) 4 (B) 4 (C) 6 (D) tn = is - 5. PQ is doule ordinte of the ellipse + 9 = 9, the norml t P meets the dimeter through Q t R, then the locus of the mid point of PR is - (A) circle (B) prol (C) n ellipse (D) hperol 6. The eqution of the chord of the ellipse + 5 = 0 which is isected t the point (, ) is - (A) = 0 (B) = (C) = 0 (D) = 7. If F & F re the feet of the perpendiculrs from the foci S & S of n ellipse n point P on the ellipse, then (S F ). (S F ) is equl to (A) (B) (C) 4 (D) 5 8. If tn. tn = ngle t - then the chord joining two points & on the ellipse on the tngent t 5 will sutend right (A) focus (B) centre (C) end of the mjor is (D) end of the minor is 9. The numer of vlues of c such tht the stright line = 4 + c touches the curve ( / 4) + = is - (A) 0 (B) (C) (D) infinite [J 98] SLCT TH CORRCT ALTRNATIVS (ON OR MOR THAN ON CORRCT ANSWRS) 0. If + k = 0 is common tngent to = 4 & ( > ), then the vlue of, k nd other common tngent re given - (A) = (B) = (C) = 0 (D) k = 4. All ellipse (0 < < ) hs fied mjor is. Tngent t n end point of ltus rectum meet t fied point which cn e - (A) (, ) (B) (0, ) (C) (0, ) (D) (0, 0). ccentric ngle of point on the ellipse + = 6 t distnce units from the centre of the ellipse is - (A) 5 (B) (C) (D) 4. For the ellipse = 0, which of the following is/re true - (A) centre is (, ) (B) length of mjor nd minor is re 8 nd 6 respectivel (C) e 7 4

15 J-Mthemtics 4. With respect to the ellipse = 8, the correct sttement(s) is/re - (A) length of ltus rectum 8 7 (B) the distnce etween the directri 7 4 (C) tngent t, is + 7 = 8 (D) Are of formed foci nd one end of minor is is On the ellipse, =, the points t which the tngents re prllel to the line 8 = 9 re - [J 99] (A), 5 5 (B), 5 5 (C), 5 5 (D), 5 5 CHCK YOUR GRASP ANSWR KY X R CI S - Que A ns. C A A B C B A B B D Que A ns. C A B B C B B B C A,B,C,D Que. 4 5 A ns. B, C A,B,D A,B,C,D A,C,D B, D 44 NOD6\\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#09\ng\0 LLIPS.p65

16 J-Mthemtics XRCIS - 0 BRAIN TASRS SLCT TH CORRCT ALTRNATIVS (ON OR MOR THAN ON CORRCT ANSWRS). + 4 = 0 is common tngent to = 4 & tngent re given -. Then the vlue of nd the other common 4 (A) ; 4 0 (B) ; 4 0 (C) ; 4 0 (D) ; 4 0. The tngent t n point P on stndrd ellipse with foci s S & S' meets the tngents t the vertices A & A' in the points V & V', then - (A) AV). A 'V ') l AV. l A 'V ' l( l( (B) (C) V 'SV 90 (D) V'S' VS is cclic qudrilterl. The re of the rectngle formed the perpendiculrs from the centre of the stndrd ellipse to the tngent nd norml t its point whose eccentric ngle is /4 is - (A) ( ) (B) ( ) (C) ( ) ( ) (D) ( ) ( ) 4. Q is point on the uilir circle of n ellipse. P is the corresponding point on ellipse. N is the foot of perpendiculr from focus S, to the tngent of uilir circle t Q. Then - (A) SP = SN (B) SP = PQ (C) PN = SP (D) NQ = SP 5. The line, l + m + n = 0 will cut the ellipse in points whose eccentric ngles differ / if - (A) l + n = m (B) m + l = n (C) l + m = n (D) n + m = l 6. A circle hs the sme centre s n ellipse & psses through the foci F & F of the ellipse, such tht the two curves intersect in 4 points. Let P e n one of their point of intersection. If the mjor is of the ellipse is 7 NOD6\\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#09\ng\0 LLIPS.p65 & the re of the tringle PF F is 0, then the distnce etween the foci is - (A) (B) (C) (D) none 7. The norml t vrile point P on n ellipse of eccentricit e meets the es of the ellipse in Q nd R then the locus of the mid-point of QR is conic with n eccentricit e' such tht - (A) e' is independent of e (B) e' = (C) e' = e (D) e' = /e 8. The length of the norml (terminted the mjor is) t point of the ellipse (A) (r + r ) (B) r r (C) where r nd r re the focl distnce of the point. + = is - rr (D) independent of r, r 45

17 J-Mthemtics 9. Point 'O' is the centre of the ellipse with mjor is AB nd minor is CD. Point F is one focus of the ellipse. If OF = 6 nd the dimeter of the inscried circle of tringle OCF is, then the product (AB)(CD) is equl to - (A) 65 (B) 5 (C) 78 (D) none 0. If P is point of the ellipse (A) PS + PS' =, if >, whose foci re S nd S'. Let PSS ' nd PS 'S, then - (B) PS + PS' =, if < (C) e tn tn (D) e when > tn tn [ ]. If the chord through the points whose eccentric ngles re on the ellipse, the focus, then the vlue of tn (/) tn (/) is - (A) e e (B) e e (C) e e psses through (D) e e. If point P( +, ) lies etween the ellipse = 0 nd its uilir circle, then - (A) [] = 0 (B) [] = (C) no such rel eist (D) [] = where [.] denotes gretest integer function.. If ltus rectum of n ellipse cosec = 5, then - 6 {0< < 4}, sutend ngle t frthest verte such tht (A) e = (B) no such ellipse eist (C) = (D) re of formed LR nd nerest verte is 6 sq. units 4. If,, s well s,, re in G.P. with the sme common rtio, then the points (, ), (, ) & (, ) - [J 99] (A) lie on stright line (B) lie on n ellipse (C) lie on circle (D) re vertices of tringle. BRAIN TASRS ANSWR KY X R CI S - Que A ns. A A,C,D A A C C C C A A,B,C Que. 4 A ns. A, B A, B A,C,D A 46 NOD6\\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#09\ng\0 LLIPS.p65

18 XRCIS - 0 FILL IN TH BLANKS J-Mthemtics. The co-ordintes of the mid - point of the vrile chord = c of the ellipse = 6 re. A tringle ABC right ngled t A moves so tht it lws circumscries the ellipse. The locus of the point A is.. Atmost normls cn e drwn from point, to n ellipse. 4. Atmost tngents cn e drwn from point, to n ellipse. MATCH TH COLUMN Following question contins sttements given in two columns, which hve to e mtched. The sttements in Column-I re lelled s A, B, C nd D while the sttements in Column-II re lelled s p, q, r nd s. An given sttement in Column-I cn hve correct mtching with ON OR MOR sttement(s) in Column-II.. Column - I Column - II (A) The minimum nd mimum distnce of point (, 6) from (p) 0 (B) (C) the ellipse re = 0 9 The minimum nd mimum distnce of point, 5 5 (q) from the ellipse 4( + 4) + 9(4 ) = 900 re If : + = nd director circle of is C, director circle of C is C director circle of C is C nd so on. If (r) 6 r, r, r... re the rdii of C, C, C... respectivel then G.M. of r, r, r is (D) Minimum re of the tringle formed n tngent to the (s) 8 ellipse + 4 = 6 with coordinte es is ASSRTION & RASON These questions contin, Sttement-I (ssertion) nd Sttement-II (reson). (A) Sttement-I is true, Sttement-II is true ; Sttement-II is correct eplntion for Sttement-I. (B) Sttement-I is true, Sttement-II is true ; Sttement-II is NOT correct eplntion for Sttement-I. (C) Sttement-I is true, Sttement-II is flse. (D) Sttement-I is flse, Sttement-II is true. MISCLLANOUS TYP QUSTIONS NOD6\\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#09\ng\0 LLIPS.p65. Sttement-I : Tngent drwn t point P 4 5, on the ellipse = 44 intersects stright line = 6 t M, then PM sutends right ngle t ( 7, 0) 7 B e c u s e Sttement-II : The portion of the tngent to n ellipse etween the point of contct nd the directri sutends right ngle t the corresponding focus. (A) A (B) B (C) C (D) D. Sttement-I : Feet of perpendiculr drwn from foci of n ellipse 4 + = 6 on the line 8 lie on circle + = 6. B e c u s e Sttement-II : If perpendiculr re drwn from foci of n ellipse to its n tngent then feet of these perpendiculr lie on director circle of the ellipse. (A) A (B) B (C) C (D) D 47

19 J-Mthemtics. Sttement-I : An chord of the ellipse + + = through (0, 0) is isected t (0, 0) B e c u s e Sttement-II : The centre of n ellipse is point through which ever chord is isected. (A) A (B) B (C) C (D) D 4. Sttement-I : If P, is point on the ellipse = 6. Circle drwn AP s dimeter touches nother circle + = 9, where A ( 5, 0) B e c u s e Sttement-II : Circle drwn with focl rdius s dimeter touches the uillir circle. (A) A (B) B (C) C (D) D COMPRHNSION BASD QUSTIONS Comprehension # : An ellipse whose distnce etween foci S nd S' is 4 units is inscried in the tringle ABC touching the sides AB, AC nd BC t P, Q nd R. If centre of ellipse is t origin nd mjor is long -is, SP + S'P = 6. On the sis of ove informtion, nswer the following questions :. If BAC = 90, then locus of point A is - (A) + = (B) + = 4 (C) + = 4 (D) none of these. If chord PQ sutends 90 ngle t centre of ellipse, then locus of A is - (A) = 60 (B) = 60 (C) = 5 (D) none of these. If difference of eccentric ngles of points P nd Q is 60, then locus of A is - (A) = 44 (B) = 576 (C) = 60 (D) = 5 MISCLLANOUS TYP QUSTION ANSWR KY X R CI S - Fill in the Blnks. 9 8 c, c 5 5 Mtch the Column. (A)(q,s) ; (B) (p,r); (C) (r); (D) (s) Assertion & Reson. D. C. A 4. A Comprehension Bsed Questions. + = +, director circle Comprehension # :. C. B. C 48 NOD6\\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#09\ng\0 LLIPS.p65

20 J-Mthemtics XRCIS - 04 [A] CONCPTUAL SUBJCTIV XRCIS. Find the eqution to the ellipse, whose focus is the point (, ), whose directri is the stright line + = 0 nd whose eccentricit is.. Find the ltus rectum, the eccentricit nd the coordintes of the foci, of the ellipse () + =, > 0 () =. Find the eccentricit of n ellipse in which distnce etween their foci is 0 nd tht of focus nd corresponding directri is If focus nd corresponding directri of n ellipse re (, 4) nd + = 0 nd eccentricit is co-ordintes of etremities of mjor is. then find the 5. An ellipse psses through the points (, ) & (, ) & its principl is re long the coordinte es in order. Find its eqution. 6. Find the ltus rectum, eccentricit, coordintes of the foci, coordintes of the vertices, the length of the es nd the centre of the ellipse = Find the set of vlue(s) of for which the point 5 7, 4 lies inside the ellipse Find the condition so tht the line p + q = r intersects the ellipse in points whose eccentric ngles differ Find the equtions of the lines with equl intercepts on the es & which touch the ellipse The tngent t the point on stndrd ellipse meets the uilir circle in two points which sutends right ngle t the centre. Show tht the eccentricit of the ellipse is ( + sin² ) /.. Find the eqution of tngents to the ellipse which psses through point (5, 4). 50 NOD6\\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#09\ng\0 LLIPS.p65. ABC is n isosceles tringle with its se BC twice its ltitude. A point P moves within the tringle such tht the squre of its distnce from BC is hlf the re of rectngle contined its distnces from the two sides. Show tht the locus of P is n ellipse with eccentricit pssing through B & C.. O is the origin & lso the centre of two concentric circles hving rdii of the inner & the outer circle s & respectivel. A line OPQ is drwn to cut the inner circle in P & the outer circle in Q. PR is drwn prllel to the -is & QR is drwn prllel to the is. Prove tht the locus of R is n ellipse touching the two circles. If the foci of this ellipse lie on the inner circle, find the rtio of inner : outer rdii & find lso the eccentricit of the ellipse. 4. If tngent drwn t point (t, t ) on the prol = 4 is sme s the norml drwn t point 5 cos, sin on the ellipse = 0, then find the vlues of t &. 5. The tngent nd norml to the ellipse + 4 = 4 t point P() on it meet the mjor is in Q nd R respectivel. If QR =, show tht the eccentric ngle of P is given cos = ± (/). 49

21 J-Mthemtics 6. If the norml t point P on the ellipse of semi es, & centre C cuts the mjor & minor es t G & g, show tht. (CG) +. (Cg) = ( ). Also prove tht CG = e CN, where PN is the ordinte of P. (N is foot of perpendiculr from P on its mjor is.) 7. A r emnting from the point ( 4, 0) is incident on the ellipse = 5 t the point P with sciss. Find the eqution of the reflected r fter first reflection. 8. Find the locus of the point the chord of contct of the tngent drwn from which to the ellipse touches the circle + = c, where c < <. 9. If + 4 = intersect the ellipse t P nd Q, then find the point of intersection of tngents t 5 6 P nd Q. 0. A tngent to the ellipse + 4 = 4 meets the ellipse + = 6 t P & Q. Prove tht the tngents t P & Q of the ellipse + = 6 re t right ngles. CONCPTUAL SUBJCTIV XRCIS ANSWR KY X R C IS - 4 ( A ) = 0. () ; 6 ; 6, 0. e , ; 5, ;, 4. ((, ) & (6, 7)) 5. ² + 5² = nd (4, ); 6 nd 4; (, ) , p + q = r sec 8 = (4 ) r = 0, = = 40, 4 5 = 00., 4. tn, t ; tn, 5 () 4 ; 5; 0, t ;, t = 48; 5 = c 5 6, 4 NOD6\\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#09\ng\0 LLIPS.p65

22 J-Mthemtics XRCIS - 04 [B] BRAIN STORMING SUBJCTIV XRCIS NOD6\\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#09\ng\0 LLIPS.p65. The tngent t n point P of circle + = meets the tngent t fied point A (, 0) in T nd T is joined to B, the other end of the dimeter through A. Prove tht the locus of the intersection of AP nd BT is n ellipse whose eccentricit is. 6. The tngent t P 4 cos, sin to the ellipse 6 + = 56 is lso tngent to the circle + 5 = 0. Find. Find lso the eqution to the common tngent.. Common tngents re drwn to the prol = 4 & the ellipse + 8 = 48 touching the prol t A & B nd the ellipse t C & D. Find the re of the qudrilterl. 4. Find the eqution of the lrgest circle with centre (, 0 ) tht cn e inscried in the ellipse + 4 = Prove tht the length of the focl chord of the ellipse t ngle is sin cos. which is inclined to the mjor is 6. The tngent t point P on the ellipse intersects the mjor is in T & N is the foot of the perpendiculr from P to the sme is. Show tht the circle on NT s dimeter intersects the uilir circle orthogonll. 7. The tngents from (, ) to the ellipse points of contct meet on the line. intersect t right ngles. Show tht the normls t the 8. If the normls t the points P, Q, R with eccentric ngles,, on the ellipse then show tht sin cos sin sin cos sin sin cos sin = 0. + = re concurrent, 9. Let d e the perpendiculr distnce from the centre of the ellipse + = to the tngent drwn t point P on the ellipse. If F & F re the two foci of the ellipse, then show tht (PF PF ) = 4 d 0. Consider the fmil of circles, + = r, < r < 5. If in the first qudrnt, the common tngent to circle of the fmil nd the ellipse = 00 meets the co-ordinte es t A & B, then find the eqution of the locus of the mid point of AB. [J 99] BRAIN STORMING SUBJCTIV XRCIS ANSWR KY X R C I S - 4 ( B ) 5. = or ; 4 ±. 55 sq. units 4. ( ) + = = 4 5

23 J-Mthemtics XRCIS - 05 [A] J-[MAIN] : PRVIOUS YAR QUSTIONS. If distnce etween the foci of n ellipse is equl to its minor is, then eccentricit of the ellipse is- [AI-00] () e = () e = () e = (4) e = 4 6. The eqution of n ellipse, whose mjor is = 8 nd eccentricit = / is- ( > ) [AI - 00 ] () + 4 = () + 4 = 48 () 4 + = 48 (4) + 9 =. The eccentricit of n ellipse, with its centre t the origin, is /. If one of the directirices is = 4, then the eqution of the ellipse is- [AI ] () + 4 = () + 4 = () 4 + = (4) 4 + = 4. An ellipse hs OB s semi minor is, F nd F' its focii nd the ngle FBF' is right ngle. Then the eccentricit of the ellipse is- 5 [AI-005, IIT-997] () () () (4) 4 5. In n ellipse, the distnce etween its foci is 6 nd minor is is 8. Then its eccentricit is- [AI ] () () 4 () (4) A focus of n ellipse is t the origin. The directri is the line = 4 nd the eccentricit is /. Then the length of the semi-mjor is is- [AI ] () 8/ () / () 4/ (4) 5/ 7. The ellipse + 4 = 4 is inscried in rectngle ligned with the coordinte es, which in turn is inscried in nother ellipse tht psses through the point (4, 0). Then the eqution of the ellipse is :- [AI ] () = 48 () = 48 () + 6 = 6 (4) + = 6 8. qution of the ellipse whose es re the es of coordintes nd which psses through the point (, ) nd hs eccentricit / 5 is :- [AI - 0 ] () = 0 () 5 + = 0 () + 5 = 0 (4) = 0 9. An ellipse is drwn tking dimeter of the circle ( ) + = s its semi-minor is nd dimeter of the circle + ( ) = 4 s its semi-mjor is. If the centre of the ellipse is t the origin nd its es re the coordinte es, then the eqution of the ellipse is : [AI - 0 ] () + 4 = 6 () 4 + = 4 () + 4 = 8 (4) 4 + = 8 0. Sttement : An eqution of common tngent to the prol = 6 nd the ellipse + = 4 is = +. Sttement : If the line = m + 4, (m 0) is common tngent to the prol m = 6 nd the ellipse + = 4, then m stisfies m 4 + m = 4. [AI - 0 ] () Sttement is true, Sttement is flse. () Sttement is flse, Sttement is true. () Sttement is true, Sttement is true ; Sttement is correct eplntion for Sttement. (4) Sttement is true, Sttement is true ; Sttement is not correct eplntion for Sttement.. The eqution of the circle pssing through the foci of the ellipse 6 9 nd hving centre t (0, ) is : [J (Min)-0] () = 0 () = 0 () = 0 (4) = 0 PRVIOUS YARS QUSTIONS ANSWR KY Q u e Ans 4 4 XRCIS-5 [A] NOD6\\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#09\ng\0 LLIPS.p65

24 J-Mthemtics XRCIS - 05 [B] J-[ADVANCD] : PRVIOUS YAR QUSTIONS. Let ABC e n equilterl tringle inscried in the circle + =.Suppose perpendiculrs from A, B, C to the mjor is of the ellipse,, ( > ) meet the ellipse respectivel t P,Q,R so tht P, Q,R lie on the sme side of the mjor is s A, B,C respectivel. Prove tht the normls to the ellipse drwn t the points P, Q nd R re concurrent. [J 000 (Mins) 7M]. Let C nd C e two circles with C ling inside C. A circle C ling inside C touches C internll nd C eternll. Identif the locus of the centre of C. [J 00 (Mins) 5M]. Prove tht, in n ellipse, the perpendiculr from focus upon n tngent nd the line joining the centre of the ellipse to the point of contct meet on the corresponding directri. [J 00 (Mins) 5M] 4. Tngent is drwn to ellipse 7 t ( cos, sin ) (where (0, /)). Then the vlue of, such tht sum of intercepts on es mde this tngent is lest is - (A) (B) 6 (C) 8 [J 00 (Screening)] 5. The re of the qudrilterl formed the tngents t the end points of the ltus rectum of the ellipse, is (A) 7/4 sq. units (B) 9 sq. units (C) 7/ sq. units (D) 7 sq. units [J 00 (Screening)] 6. Find point on the curve + = 6 whose distnce from the line + = 7, is s smll s possile. [J 00 (Min) M out of 60] 7. Locus of the mid points of the segments which re tngents to the ellipse intercepted etween the coordinte es is - (A) (B) 4 (D) 4 nd which re [J 004 (Screening)] 4 (C) 4 (D) 4 NOD6\\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#09\ng\0 LLIPS.p65 8. The minimum re of tringle formed tngent to the ellipse (A) (B) (C) ( ) nd coordinte es - (D) [J 005 (Screening)] 9. Find the eqution of the common tngent in st qudrnt to the circle + = 6 nd the ellipse. 5 4 Also find the length of the intercept of the tngent etween the coordinte es. [J 005 (Mins) 4M out of 60] 0. Let P(, ) nd Q(, ), < 0, < 0, e the end points of the ltus rectum of the ellipse +4 =4. The equtions of prols with ltus rectum PQ re - [J 008, 4M] (A) + = + (B) = + (C) + = (D) = 5

25 J-Mthemtics. The line pssing through the etremit A of the mjor is nd etremit B of the minor is of the ellipse + 9 = 9 meets its uilir circle t the point M. Then the re of the tringle with vertices t A, M nd the origin O is :- (A) 0 (B) 9 0 (C) 0 (D) 7 0 [J 009, M, M]. The norml t point P on the ellipse + 4 = 6 meets the -is t Q. If M is the mid point of the line segment PQ, then the locus of M intersects the ltus rectums of the given ellipse t the points - [J 009, M, M] (A) 5, 7 (B) Prgrph for Question to Tngents re drwn from the point P(, 4) to the ellipse B.. The coordintes of A nd B re (A) (, 0) nd (0, ) (C) 8 6, 5 5 nd (0, ) 4. The orthocenter of the tringle PAB is (A) 8 5, 7 (B) 7 5, 5 8 (C) (B), 7 (D) 4, 7 [J 0, (M ech), M] touching the ellipse t points A nd , 5 5 (D) (, 0) nd (C) 8, 5 5 nd 9 8, , 5 5 (D) 8 7, The eqution of the locus of the point whose distnces from the point P nd the line AB re equl, is (A) = 0 (B) = 0 (C) = 0 (D) = 0 6. The ellipse : 9 4 is inscried in rectngle R whose sides re prllel to the coordinte es. Another ellipse pssing through the point (0,4) circumscries the rectngle R. The eccentricit of the ellipse is - [J 0, M, M] (A) (B) 54 (C) 7. A verticl line pssing through the point (h,0) intersects the ellipse 4 (D) 4 the tngents to the ellipse t P nd Q meet t the point R. If (h) = re of the tringle PQR, nd 8 min (h), then 8 5 / h PRVIOUS YARS QUSTIONS t the points P nd Q. Let m (h) / h [J- Advnced 0, 4, ( )M]. Locus is n ellipse with foci s the centres of the circles C nd C. 4. B 5. D 6. (,) 7. D 8. A C ; 4 ANSWR KY XRCIS-5 [B] 0. B,C. D. C. D 4. C 5. A NOD6\\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#09\ng\0 LLIPS.p65