HYPERBOLA AIEEE Sllus. Stndrd eqution nd definitions. Conjugte Hperol. Prmetric eqution of te Hperol. Position of point P(, ) wit respect to Hperol 5. Line nd Hperol 6. Eqution of te Tngent Totl No. of questions in Ellipse re: Solved emples..... 0 Level #. Level #.. Level #.. Totl No. of questions....88. Students re dvised to solve te questions of eercises (Levels #,,, ) in te sme sequence or s directed te fcult memers.. Level # is not for foundtion course students, it will e discussed in freser nd trget courses. IIT-JEE PREPRETION MATHE HYPERBOLA
Inde : Prepring our own list of Importnt/Difficult Questions Instruction to fill (A) Write down te Question Numer ou re unle to solve in column A elow, Pen. (B) After discussing te Questions written in column A wit fculties, strike off tem in te mnner so tt ou cn see t te time of Revision lso, to solve tese questions gin. Write down te Question Numer ou feel re importnt or good in te column B. EXERCISE NO. COLUMN :A COLUMN :B Questions I m unle to solve in first ttempt Good/Importnt questions Level # Level # Level # Level # Advntges. It is dvised to te students tt te sould prepre question nk for te revision s it is ver difficult to solve ll te questions t te time of revision.. Using ove indeou cn prepre nd mintin te questions for our revision. IIT-JEE PREPRETION MATHE HYPERBOLA
. Stndrd Eqution nd Definitions KEY CONCEPTS Stndrd Eqution of perol is z z T M M L P(, ) S ( e, 0) K A(, 0) (0, 0) O K A (, 0) S (e, 0) T L (i) Definition perol : A Hperol is te locus of point in plne wic moves in te plne in suc w tt te rtio of its distnce from fied point (clled focus) in te sme plne to its distnce from fied line (clled directri) is lws constnt wic is lws greter tn unit. (ii) Vertices : Te point A nd A were te curve meets te line joining te foci S nd S re clled vertices of perol. (iii) Trnsverse nd Conjugte es : Te strigt line joining te vertices A nd A is clled trnsverse es of te perol. Strigt line perpendiculr to te trnsverse es nd psses troug its centre clled conjugte es. (iv) Ltus Rectum : Te cord of te perol wic psses troug te focus nd is perpendiculr to its trnsverse es is clled ltus rectum. Lengt of ltus rectum = (v) Eccentricit :. For te perol e = = (vi) Focl distnce :, = (e ) Conjugte es Trnsverse es Te distnce of n point on te perol from te focus is clled te focl distnce of te point. Note : Te difference of te focl distnce of point on te perol is constnt nd is equl to te lengt of te trnsverse es. SP SP = (const.). Conjugte Hperol Note : Te perol wose trnsverse nd conjugte es re respectivel te conjugte nd trnsverse es of given perol is clled conjugte perol. Eqution of conjugte perol + (i) If e nd e re te eccentricities of te perol nd its conjugte ten + e e IIT-JEE PREPRETION MATHE HYPERBOLA
(ii) Te focus of perol nd its conjugte re concclic. S.No. Prticulrs Hperol Conjugte Hperol +. Co-ordinte of te centre (0, 0) (0, 0). Co-ordinte of te vertices (, 0) & (, 0) (0, ) & (0, ). Co-ordinte of foci ( ± e, 0) (0, ± e). Lengt of te trnsverse es 5. Lengt of te conjugte es 6. Eqution of directri = ± /e = ± /e 7. Eccentricit e = e = 8. Lengt of ltus rectum. Eqution of trnsverse es = 0 = 0 0. Eqution of conjugte es = 0 = 0. Prmetric eqution of te Hperol Let te eqution of ellipse in stndrd form will e given Ten te eqution of ellipse in te prmetric form will e given = sec, = tn were is te eccentric ngle wose vlue vr from 0 <. Terefore coordinte of n point P on te ellipse will e given ( sec, tn ).. Position of point P(, ) wit respect to Hperol Te quntit is positive, zero or negtive ccording s te point (, ) lies inside on or outside te perol 5. Line nd Hperol. Te strigt line = m + c is scnt, tngent or psses outside te perol s c > = < m 6. Eqution of Tngent ccording (i) Te eqution of tngents of slope m to te perol re = m + m nd te co-ordintes of te point of contcts re m m, m (ii) Eqution of tngent to te perol t te point (, ) is (iii) Eqution of tngent to te perol t te point ( sec, tn) is sec tn IIT-JEE PREPRETION MATHE HYPERBOLA 5
Note : E. In generl two tngents cn e drwn from n eternl point (, ) to te perol nd te re = m ( ) nd = m ( ), were m nd m re roots of Find te eqution of te perol wose directri is +, focus (, ) nd eccentricit. Sol. Let P (,) e n point on te perol. Drw PM perpendiculr from P on te directri. Ten definition SP = e PM (SP) = e (PM) ( ) + ( ) = 5( + + 5} = ( + + + ) 7 + + = 0 wic is te required perol. E. Find te lengts of trnsverse is nd conjugte is, eccentricit nd te coordintes of foci nd vertices; lengts of te ltus rectum, equtions of te directrices of te perol 6. Sol. Te eqution 6 = cn e written s =. Tis is of te form 6 = =, 6 =, = Lengt of trnsverse is : Te lengt of trnsverse is = = 8 Lengt of conjugte is : Te lengt of conjugte is = = 6 Eccentricit : e = = SOLVED EXAMPLES 5 = 6 Foci : te co- ordintes of te foci re (0,± e), i.e., (0, ± ) Lengt of Ltus rectum : Te lengt of ltus rectum = Eqution of directrices : = () = ( ) m + + = 0 Te eqution of directrices re = ± e IIT-JEE PREPRETION MATHE HYPERBOLA 5 = ± (5/ ) 6 = ± 5 E. Find te position of te point (5, ) reltive to te perol. Sol. Since (5) () = 5 6 = 08 > 0 so te point (5, ) lies outside te perol E. Te line 5 + = touces te perol = t te point (A) ( 5,/) (B) (5, /) (, /) 5 Sol.[B] We ve : m = Slope of te tngent = E. 5 If line of slope m is tngent to te perol, ten te coordintes of te point of contct re m, m m Here, =, nd m = 5/ So, points of contct re 5, i.e. 5, nd 5,. Out of tese two points 5, lies on te line 5 + =. Hence, 5, is te required point. Te eqution of te common tngents to te prol = 8 nd te perol = is - (A) ± + = 0 (B) ± + = 0 ± + = 0 (D) ± + = 0 Sol.[A] Prol = 8 = 8 = An tngent to te prol is = m + m...(i)
E.6 If it is lso tngent to te perol i.e., = ten c = m.m m or m m = 0 (m ) (m + ) = 0 m = ± putting for m in (i), we get te tngents s ± + = 0 Te locus of te point of intersection of te lines k = 0 nd k + k = 0 for different vlues of k is - (A) Ellipse (B) Prol Circle (D) Hperol Sol.[D] = k...(i) E.7 nd k + k = 0 k ( + ) =...(ii) To find te locus of teir point of intersection eliminte te vrile K etween te equtions from (i) K = nd putting in (ii), we get ( ) ( + ) = ( ) = 8 or 6 8 Hence te locus is perol Te eccentricit of te conic represented + + 6 = 0 is - (A) (B) (D) Sol.[B] We ve + + 6 = 0 or ( ) ( ) = 6 or ( + ) ( + ) = 6 or ( ) ( ) = 6 ( ) ( ) or = i.e. e + ( conjugte perol) e + e = E.8 Te eqution 6 8 + 5 = 0 represent perol - (A) Te lengt of te trnsverse es is (B) Lengt of ltus rectum is Eqution of directri is = nd = 5 5 Sol.[C] We ve 6 8 + 5 = 0 ( ) 6( ) 5 ( +) 6( + ) ( ) 6( ) ( ) 6 = ±7 or + ± = 0 IIT-JEE PREPRETION MATHE HYPERBOLA 7 ( ) X Y Compring wit were X =, Y = nd 6, = so Te lengt of te trnsverse es = = 8 Te lengt of te ltus rectum = = Te eqution of te directri X = ± e 6 6 = ± = ± + 5 5 = ; = 5 5 E. For wt vlue of does te line = + touces te perol 6? Sol. Eqution of perol is 6 E.0 Sol. or = compring tis wit 6, we get =, 6 nd compring tis line = + wit = m + c ; m = & c = If te line = + touces te perol 6 ten c = m = () 6 = ±5 Find te eqution of te tngent to te perol = 6 wic is perpendiculr to te line + = 0. Let m e te slope of te tngent since te tngent is perpendiculr to te line + = 0. m m since = 6 or 6 Compring tis wit ; = 6 & = so te eqution of tngents re = ( ) ± 6 ( )
Question sed on Eqution & Properties of Hperol Q. Te vertices of perol re t (0, 0) nd (0, 0) nd one of its foci is t (8, 0). Te eqution of te perol is - (A) 5 ( 5) (B) 5 ( 5) 5 ( 5) ( 5) (D) 5 Q. If te ltus rectum of n perol e 8 nd eccentricit e 5, ten te eqution of te perol is- (A) 5 00 (B) 5 00 + 5 00 (D) 5 + 00 LEVEL- Q.7 Eqution of te perol wit eccentricit / nd foci t (±, 0) is- (A) = 5 (B) = Q.8 If te centre, verte nd focus of perol e (0, 0), (, 0) nd (6, 0) respectivel, ten te eqution of te perol is- (A) 5 = 8 (B) 5 = 80 5 = 80 (D) 5 = 8 Q. Te eccentricit of te perol cn never e equl to- (A) 5 (B) (D) 8 Q. Te foci of te perol 6 + 8 + 5 = 0 re- (A) (, ), (5, 7) (B) (, ), (6, ) (0, 0), (5, ) Q. Te foci of te perol 6 = 0 re- (A) [±, 0] (B) [±, 0] [±, 0] (D) [0, ± ] Q.5 Foci of te perol 6 ( ) re (A) (5, ); (5, ) (B) (5, ); (5, ) (5, ); (5, ) Q.6 Te eccentricit of perol pssing troug te points (, 0), (, ) will e- (A) (B) (D) Q.0 Te eccentricit of te perol wose ltus rectum is 8 nd conjugte is is equl to lf te distnce etween te foci is- (A) (B) Q. If te lengt of te trnsverse nd conjugte es of perol e 8 nd 6 respectivel, ten te difference of focl distnces of n point of te perol will e- (A) 8 (B) 6 (D) Q. If m is vrile, te locus of te point of intersection of te lines + = m (A) prol perol is / n- = m nd (B) ellipse IIT-JEE PREPRETION MATHE HYPERBOLA 8
Q. Te eqution of te perol wose foci re (6, 5), (, 5) nd eccentricit 5/ is- (A) ( ) 6 (B) 6 ( ) Q. Te eqution ( 5) ( 5) 6 + (A) perol if < 8 (B) n ellipse if > 8 8 perol if 8 < < represents Q.5 Te eqution + + + g + f + c = 0 represents rectngulr perol if- (A) 0, >, + = 0 (B) 0, <, + = 0 0, =, + = 0 Q.6 Te eqution k, k > represents- k (A) circle (B) ellipse perol Q.7 If e nd e e te eccentricities of two conics S nd S suc tt e + e =, ten ot S nd S re- (A) ellipse perols (B) prols Q.8 A point moves in plne so tt its distnces PA nd PB from two fied points A nd B in te plne stisf te reltion PA PB = k (k 0), ten te locus of P is- (A) prol (B) n ellipse perol (D) rnc of perol Q. Te eqution of te conic wit focus t (, ), directri long + = 0 nd wit eccentricit is- (A) (B) + + = 0 (D) + = 0 Q.0 Te lengt of te ltus rectum of te perol (A) is- (B) (D) Q. Te eqution 6 + = 0 represents perol- (A) te lengt of wose trnsverse is is (B) te lengt of wose conjugte is is wose centre is (, ) (D) wose eccentricit is Q. Te lengt of te trnsverse is of perol is 7 nd it psses troug te point (5, ). Te eqution of te perol is- (A) (B) 6 5 5 6 5 6 (D) none of tese Q. Te ltus rectum of perol 6. Its eccentricit e = (A) /5 (B) 5/ / (D) / = is p Q. Consider te set of perol = k, k R. Let e e te eccentricit wen k = nd e e te eccentricit wen k =. Ten e + e = (A) (B) (D) IIT-JEE PREPRETION MATHE HYPERBOLA
Q.5 Te eccentricit of te perol + is given - (A) e = + e = + (B) e = + (D) e = + Q.6 If e nd e' e te eccentricities of perol Question sed on nd its conjugte, ten + = e e (A) 0 (B) Line nd Hperol Q.7 Te eqution of tngent prllel to = drwn to is- (A) + = 0 (B) + = 0 + = 0 (D) + = 0 Q.8 Te line l + m + n = 0 will e tngent to te perol, if - (A) l + m = n (B) l m = n m n = l Q. Te eqution of tngents to te perol = 6 wic re perpendiculr to te line + = 0 (A) = + (B) = = ± Q.0 Te line = + touces te perol 5 = 5 t te point- (A) (0, ) (B) (, ) (/, 5/) Q. Eqution of tngent to te perol = 6 wic is prllel to te line = + is- (A) = + 5 (B) = 5 = + 5 nd = 5 (D) none of tese Q. If te strigt line cos + sin = p e tngent to te perol (A) cos + sin = p (B) cos sin = p sin + cos = p (D) sin cos = p, ten Q. Te vlue of m for wic = m + 6 is tngent to te perol is- 00 (A) 7 0 0 (B) (D) 0 7 0 Q. Eqution of one of common tngent to prol = 8 nd perol = is- (A) = 0 (B) + = 0 + + = 0 (D) + = 0 IIT-JEE PREPRETION MATHE HYPERBOLA 0
LEVEL- Q. Te ltus rectum sutends rigt ngle t oter focus of perol ten its eccentricit is- (A) + (B) + + (D) Q. Te eqution of te perol wose foci re te foci of te ellipse nd te 5 eccentricit is, is - (A) (B) (D) Q. A tngent to perol intercepts lengt of unit from ec of te coordinte es, ten te point (, ) lies on te- (A) = (B) (D) none of tese Q. A common tngent to 6 nd + = is - (A) = 5 + 7 7 5 (B) = + 7 7 = + 5 7 (D) none of tese 7 Q.5 Te product of te lengts of te perpendiculrs drwn from foci on n tngent to te perol is - (A) (B) (D) / Q.6 Te re of qudrilterl formed focii of perol nd its conjugte perol is- (A) (B) Q.7 Te equtions to te common tngents to te two perols re- (A) = ± ± (B) = ± ± = ± ± ( ) (D) = ± ± nd Q.8 A perol s es long coordinte es. Its trnsverse is is nd it psses troug (,k) ten its eccentricit is- (A) k k Q. If te focii of te ellipse te perol vlue of k = (B) (D) (A) ± (B) ± (D) k k k + nd coincides ten Q.0 Te locus of te point of intersection of te lines t t = nd + = t is - (A) prol perol (B) n ellipse Q. Let LL e te ltus rectum troug te focus S of perol nd A e te frter verte of te conic. If ALL is equilterl ten its eccentricit e = (A) (B) + ( + )/ (D) ( + )/ IIT-JEE PREPRETION MATHE HYPERBOLA
Q. If te ltus rectum sutends rigt ngle t te centre of te perol ten its eccentricit is (A) e = ( )/ (B) e = ( 5 ) / e = ( 5 + )/ (D) e = ( + )/ Q. Te eqution = represents (A) n ellipse perol t t e e ; = t t e e (B) prol (D) circle ; t R Q. If te tngent t te point ( sec, tn ) of te perol is prllel to + = 0, ten te vlue of is- (A) 5 (B) 60 0 (D) 75 Q.5 Te ellipse + nd te perol 5 6 ve in common- 5 6 (A) centre onl (B) centre, foci nd directrices centre, foci nd vertices (D) centre nd vertices onl Q.6 Te tngents to te perol = re prllel to te strigt line + + 8 = 0 t te following points- (A) (, ) (B) (, ) (, ) (D) (, ) Q.7 P is point on te perol, N is te foot of te perpendiculr from P on te trnsverse is. Te tngent to te perol t P meets te trnsverse is t T. If O is te centre of te perol, ten OT.ON is equl to - (A) e (B) (D) Q.8 If e, e re te eccentricities of te ellipse + nd te perol 8 respectivel, ten te reltion etween e nd e is - (A) e + e = (B) e + e = e + e = (D) e + e = Q. Te vlue of m for wic line = m + 5 touces te perol 6 re te roots of te eqution ( + ) = 0, ten ( + ) is equl to- (A) (B) 0 (D) none of tese Q.0 Te re of tringle formed lines = 0 nd n tngent to te perol = is- (A) (B) Q. If te distnces etween te foci nd te distnce etween te directrices of te perol re in te rtio : ten : is- (A) : (B) : : (D) : Q. If te eccentricit of te perol sec = 5 is times te eccentricit of te ellipse sec + = 5, ten vlue of is- (A) 6 (B) (D) Q. If te tngent t te point P( sec, tn ) to te perol meets its trnsverse is t T, ten ST (S focus) must e equl to (A) (e cos ) (B) (e + cos) (e + cos ) (D) e cot Assertion-Reson: (Q. No. to 7) Te following questions given elow consist of n Assertion () nd Reson () Tpe questions. Use te following ke to coose te pproprite nswer. (A) Bot () nd () re true nd () is te correct eplntion of () (B) Bot () nd () re true ut () is not te correct eplntion of () () is true ut () is flse (D) () is flse ut () is true IIT-JEE PREPRETION MATHE HYPERBOLA
Q. Sttement-(): If P(, ) is point on + = ten re SPS = e Sttement-() : A tngent to meets te trnsverse nd conjugte is in P nd Q ten CP CQ, were C is te centre of te conic. Wic of te sttements is correct? (A) ot nd (B) onl onl (D) neiter nor Q.5 Sttement-() : Te conic 6 + = 0 represent perol. Sttement- () : Te squre of te coefficient of is greter tn te product of te coefficient of & nd 0. Q.6 Sttement-(): Te ltus-rectum of te perol = is equl to te lengt of its trnsverse is. Sttement-(): Te semi ltusrectum of te ellipse + = is equl to Q.7 Sttement- () : Te eqution. + + + + + = 0 cn never represent perol. Sttement- (): Te generl eqution of second degree represents perol if >. Pssge : (Q.No.8 & ) If prmetric eqution of perol is = t t e e & = t t e e Q.8 Eccentricit of perol is (A) (B) ten. (D) Q. Eccentric ngle of point, on perol (A) 6 (B) Column Mtcing Questions: (Q.No.0 to ) Q.0 Mtc te column Column I Column II (A) Te numer of tngents (P) from (, ) to te perol 6 (B) Te numer of tngents (Q) from (, ) to te ellipse 6 + If e nd e re te (R) eccentricities of = c nd = ten e + e = (S) 0 (T) Q. Mtc te following curves wit teir respective prmeter points: Column I Column II (A) (P) t, t (B) = c c (Q) ct, t (R) ( sec t, tn t) (S) (c tn t, c cot t) (T) ( cos t, sin t) t t IIT-JEE PREPRETION MATHE HYPERBOLA
SECTION -A LEVEL- (Question sked in previous AIEEE nd IIT-JEE) Q. Te ltus rectum of te perol [IIT Scr. 00] 6 is- [AIEEE-00] (A) 6/ (B) / 8/ (D) / Q. Te point t wic te line + 6 = touces te curve =, is- (A) (, 6 ) (B) ( 6, ), (D) 6, 6 Q. Te foci of te ellipse + 6 nd te perol = coincide. Ten te 8 5 vlue of is- [AIEEE- 00] (A) (B) 5 (D) 7 Q. Te locus of point P(, ) moving under te condition tt te line = + is tngent to te perol (A) n ellipse prol is- [AIEEE-005] (B) circle (D) perol Q. Te eqution of te perol wose foci re (, 0) nd (, 0) nd eccentricit is is given - SECTION -B Q. If [AIEEE-0] (A) = (B) = + = (D) + = cos represents fmil of sin perols, were vries ten- (A) e remins constnt [IIT Scr.00/AIEEE-07] (B) scisss of foci remin constnt eqution of directrices remin constnt (D) sciss of vertices remin constnt Q. If perol psses troug te focus of te + nd its trnsverse nd conjugte 5 6 es coincide wit te mjor nd minor is of ellipse, nd product of eccentricities is, ten (A) Focus of perol is (5, 0) (B) Focus of perol is (5, 0) Te eqution of perol is (D) Te eqution of perol is [IIT-006] 5 6 Q. A perol, ving te trnsverse is of lengt sin, is confocl wit te ellipse +. Ten its eqution is- [IIT-007] (A) cosec sec (B) sec cosec sin cos (D) cos sin Q.5 An ellipse intersects te perol ortogonll. Te eccentricit of te ellipse is reciprocl of tt of te perol. If te es of te ellipse re long te coordinte es, ten (A) eqution of ellipse is + = (B) te foci of ellipse re (±, 0) eqution of ellipse is + = (D) te foci of ellipse re (±, 0) [IIT-00] IIT-JEE PREPRETION MATHE HYPERBOLA
Q.6 Te line + is tngent to te perol =. If tis line psses troug te point of intersection of te nerest directri nd te -is, ten te eccentricit of te perol is - [IIT-00] (A) (B) (D) Pssge : (Q.7 to Q.8) Te circle + 8 = 0 nd perol intersect t te points A nd B [IIT-00] Q.7 Eqution of common tngent wit positive slope to te circle s well s to te perol is - (A) 5 0 = 0 (B) 5 + = 0 + 8 = 0 (D) + = 0 Q. Let te eccentricit of te perol e reciprocl to tt of te ellipse + =. If te perol psses troug focus of te ellipse, ten - (A) te eqution of te perol is (B) focus of te perol is (, 0) te eccentricit of te perol is [IIT-0] (D) te eqution of te perol is = 5 Q.8 Eqution of te circle wit AB s its dimeter is (A) + + = 0 (B) + + + = 0 + + = 0 (D) + = 0 IIT-JEE PREPRETION MATHE HYPERBOLA 5
ANSWER KEY LEVEL- Q.No. 5 6 7 8 0 Ans. B A B C A B A C B C A C A Q.No. 5 6 7 8 0 5 6 Ans. C A D C C C A D C B C D B Q.No. 7 8 0 Ans. A B A C C B A B,C LEVEL- Q.No. 5 6 7 8 0 Ans. B B B B B A B C A C D C C Q.No. 5 6 7 8 0 5 6 Ans. C D B,C B C C C A B A,C C A A Q.No. 7 8 Ans. D B C (0) A T ; B S ; C Q () A T ; B Q, S ; C P, R LEVEL- SECTION-A Qus. Ans. B D D B SECTION-B Qus. 5 6 7 8 Ans. B A A,D A A,B A B A B,D IIT-JEE PREPRETION MATHE HYPERBOLA 6