Ellipse. 1. Defini t ions. FREE Download Study Package from website: 11 of 91CONIC SECTION
|
|
- Patience Antonia Small
- 6 years ago
- Views:
Transcription
1 FREE Downlod Stud Pckge from wesite: Defini t ions Ellipse It is locus of point which moves in such w tht the rtio of its distnce from fied point nd fied line (not psses through fied point nd ll points nd line lies in sme plne) is constnt (e) which is less thn one. The fied point is clled - focus The fied line is clled -directri. The constnt rtio is clled - eccentricit, it is denoted 'e'. Solved Emple # Find the eqution to the ellipse whose focus is the point (, ), whose directri is the stright line 3 0 nd eccentricit is. Let P (h, k) e moving point, PS e PM h k 3 (h ) (k ) 4 locus of P(h, k) is 8 { } ( 6 6 9) Note : The generl eqution of conic with focus (p, q) & directri l m n 0 is: (l m ) [( p) ( q) ] e (l m n) h g f c 0 represent ellipse if 0 < e < ; 0, h² < Self Prctice Prolem :-. Find the eqution to the ellipse whose focus is (0, 0) directri is 0 nd e Stndrd Eqution Stndrd eqution of n ellipse referred to its principl es long the co ordinte es is, where > & ² ² ( e²). Eccentricit: e, (0 < e < ) Focii : S ( e, 0) & S ( e, 0). Equtions of Directrices : e & e. Mjor Ais : The line segment A A in which the focii 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). Minor Ais : The is intersects the ellipse in the points B (0, ) & B (0, ). The line segment B B is of length ( < ) is clled the minor is of the ellipse. Principl Ais : The mjor & minor es together re clled principl is of the ellipse. Vertices : Point of intersection of ellipse with mjor is. A (, 0) & A (, 0). Focl Chord : A chord which psses through focus is clled focl chord. Doule Ordinte : A chord perpendiculr to the mjor is is clled doule ordinte. Ltus Rectum : The focl chord perpendiculr to the mjor is is clled the ltus rectum. Length of ltus rectum (LL ) ( minor is) ( e ) mjor is. of 9CONIC SECTION TEKO CLASSES, H.O.D. MATHS : SUHAG R. KARIYA (S. R. K. Sir) PH: (0755) , , BHOPAL, (M.P.)
2 FREE Downlod Stud Pckge from wesite: e (distnce from focus to the corresponding directri) 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. NOTE : (i) If the eqution of the ellipse is given s nd nothing is mentioned, then the rule is to ssume tht >. (ii) If > is given, then the is will ecome mjor is nd -is will ecome the minor is nd ll other points nd lines will chnge ccordingl. Solved Emple # : Find the eqution to the ellipse whose centre is origin, es re the es of co-ordinte nd psses through the points (, ) nd (3, ). Let the eqution to the ellipse is Since it psses through the points (, ) nd (3, ) (i) 9 nd...(ii) from (i) 4 (ii), we get from (i), we get Solved Emple # 3 Ellipse is Find the eqution of the ellipse whose focii re (4, 0) nd ( 4, 0) nd eccentricit is 3 Since oth focus lies on -is, therefore -is is mjor is nd mid point of focii is origin which is centre nd line perpendiculr to mjor is nd psses throguh centre is minor is which is -is. Let eqution of ellipse is e 4 nd e 3 (Given) nd ( e ) Eqution of ellipse is 44 8 Solved Emple # 4 If minor-is of ellipse sutend right ngle t its focus then find the eccentricit of ellipse. Let the eqution of ellipse is π BSB nd OB OB BSO 4 π OS OB e ( > ) of 9CONIC SECTION TEKO CLASSES, H.O.D. MATHS : SUHAG R. KARIYA (S. R. K. Sir) PH: (0755) , , BHOPAL, (M.P.)
3 FREE Downlod Stud Pckge from wesite: e e e Solved Emple # 5: From point Q on the circle, perpendiculr QM re drwn to -is, find the locus of point 'P' dividing QM in rtio :. Let Q ( cosθ, sinθ) M ( cosθ, 0) Let P (h, k) sinθ h cosθ, k 3 3k h Locus of P is ( /3) Solved Emple # 6 Find the eqution of es, directri, co-ordinte of focii, centre, vertices, length of ltus - rectum nd eccentricit of n ellipse ( 3) 5 ( ) 6 X Y Let 3 X, Y, so eqution of ellipse ecomes s 5 4 eqution of mjor is is Y 0.. eqution of minor is is X 0 3. centre (X 0, Y 0) 3, C (3, ) Length of semi-mjor is 5 Length of mjor is 0 Length of semi-minor is 4 Length of mjor is 8. Let 'e' e eccentricit ( e ) e Length of ltus rectum LL Co-ordintes focii re X ± e, Y 0 S (X 3, Y 0) & S (X 3, Y 0) S (6, ) & S (0, ) Co-ordinte of vertices Etremities of mjor is A (X, Y 0) & A (X, Y 0) A ( 8, ) & A (, ) A (8, ) & A (, ) Etremities of minor is B (X 0, Y ) & B (X 0, Y ) B ( 3, 6) & B ( 3, ) B (3, 6) & B (3, ) Eqution of directri X ± e 5 3 ± & 3 Self Prctice Prolem. Find the eqution to the ellipse whose es re of lengths 6 nd 6 nd their equtions re nd 3 0 respectivel. 3( 3 3) (3 ) 80, Find the eccentricit of ellipse whose minor is is doule the ltus rectum. 3 3 of 9CONIC SECTION TEKO CLASSES, H.O.D. MATHS : SUHAG R. KARIYA (S. R. K. Sir) PH: (0755) , , BHOPAL, (M.P.)
4 FREE Downlod Stud Pckge from wesite: 4. Find the co-ordintes of the focii of the ellipse ±, Find the stndrd ellipse pssing through (, ) nd hving eccentricit. 6. A point moves so tht the sum of the squres of its distnces from two intersecting non perpendiculr stright lines is constnt. Prove tht its locus is n ellipse. 3. Auilir Circle / Eccentric Angle : A circle descried on mjor is of ellipse s dimeter is clled the uilir circle. Let Q e point on the uilir circle ² ² ² such tht line through Q perpendiculr to the is on the w intersects the ellipse t P, then P & Q re clled s the Corresponding Points on the ellipse & the uilir circle respectivel. θ is clled the Eccentric Angle of the point P on the ellipse ( π < θ π). Q ( cosθ, sinθ) P ( cosθ, sinθ) Note tht : l(pn) Semi minor is l(qn) Semi mjor is NOTE : If f rom ech point of circl e 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. Solved Emple # 7 Find the focl distnce of point P(θ) on the ellipse Let 'e' e the eccentricit of ellipse. PS e. PM nd e cosθ e PS ( e cosθ) PS e. PM e cos θ e PS e cosθ focl distnce re ( ± e cosθ) Note : PS PS PS PS AA Solve Emple # 8 Find the distnce from centre of the point P on the ellipse with -is. Sol. Let P ( cosθ, sinθ) tnθ tnα tnθ OP tn α cos θ sin θ tn θ tn θ sec θ tn θ tn tn α α ( > ) whose rdius mkes ngle α 4 of 9CONIC SECTION TEKO CLASSES, H.O.D. MATHS : SUHAG R. KARIYA (S. R. K. Sir) PH: (0755) , , BHOPAL, (M.P.)
5 FREE Downlod Stud Pckge from wesite: OP Self Prctice Prolem sin α cos α 7. Find the distnce from centre of the point P on the ellipse r cos α sin α whose eccentine ngle is α 8. Find the eccentric ngle of point on the ellipse whose distnce from the centre is. 6 π 3π ±, ± Show tht the re of tringle inscried in n ellipse ers constnt rtio to the re of the tringle formed joining points on the uilir circle corresponding to the vertices of the first tringle. 4. Prm etric Representtio n: The equtions cos θ & sin θ together represent the ellipse. Where θ is prmeter. Note tht if P(θ) ( cos θ, sin θ) is on the ellipse then; Q(θ) ( cos θ, sin θ) is on the uilir circle. The eqution to the chord of the ellipse joining two points with eccentric ngles α & β is given α β α β α β cos sin cos. Solved Emple # 9 Write the eqution of chord of n ellipse joining two points P 5 6 Eqution of chord is π 5π π 5π cos 4 4. sin 4 4 cos 5 4. cos 5 3π 4 3π. sin π 5π 4 4 π 4 nd Q 5π. 4 If P(α) nd P(β) re etremities of focl chord of ellipse then prove tht its eccentricit e α β cos α β cos Let the eqution of ellipse is eqution of chord is. α β α β α β cos sin cos Since ove chord is focl chord, it psses through focus (e, 0) or ( e, 0) α β α β ± e cos cos e α β cos α β cos 5 of 9CONIC SECTION TEKO CLASSES, H.O.D. MATHS : SUHAG R. KARIYA (S. R. K. Sir) PH: (0755) , , BHOPAL, (M.P.)
6 Note : ± e α β cos α β cos α β tn. tn ± e α β tn. tn Appling componendo nd dividendo 6 of 9CONIC SECTION FREE Downlod Stud Pckge from wesite: ± e ± e α β tn. tn α β e tn tn or e Solved Emple # e e Find the ngle etween two dimeters of the ellipse ngle α nd β α π. Let ellipse is sinα Slope of OP m tnα cos α sinβ π Slope of OQ m cotα given β α cosβ tnθ Self Prctice Prolem m m m m (tn α cot α) 0. Find the sum of squres of two dimeters of the ellipse eccentric ngles differ π nd show tht it is constnt. 4( ). Whose etremities hve eccentrici ( )sinα whose etremitites hve. Show tht the sum of squres of reciprocls of two perpendiculr dimeters of the ellipse is constnt. Find the constnt lso. 4. Find the locus of the foot of the perpendiculr from the centre of the ellipse joining two points whose eccentric ngles differ π. ( ). 5. Position of Point w.r.t. n Ellipse: on the chord The point P(, ) lies outside, inside or on the ellipse ccording s ; > < or 0. Solved Emple # TEKO CLASSES, H.O.D. MATHS : SUHAG R. KARIYA (S. R. K. Sir) PH: (0755) , , BHOPAL, (M.P.) Check wether the point P(3, ) lies inside or outside of the ellipse. 5 6
7 FREE Downlod Stud Pckge from wesite: S < 0 Solved Emple # 3 Point P (3, ) lies inside the ellipse. Find the set of vlue(s) of 'α' for which the point P(α, α) lies inside the ellipse 6 9. If P(α, α) lies inside the ellipse S < 0 α α < α < α < 44 5 α Solved Emple 0, Line nd n Ellipse: The line m c meets the ellipse in two points rel, coincident or imginr ccording s c² is < or > ²m² ². Hence m c is tngent to the ellipse if c² ²m² ². Solved Emple # 4 Find the set of vlue(s) of 'λ' for which the line 3 4 λ 0 intersect the ellipse 6 9 t two distinct points. Solution Solving given line with ellipse, we get (4 λ) λ λ Since, line intersect the prol t two distinct points, roots of ove eqution re rel & distinct D > 0 λ (8) 8. 9 λ 44 > 0 < λ < Self Prctice Prolem 3. Find the vlue of 'λ' for which λ 0 touches the ellipse 5 9 λ ± 09 m is tngent to the ellipse vlues of m. () Point form : is tngent to the ellipse t (, ). 7. Tngent s:() Slope form: m ± for ll (c) Prmetric form: cosθ sinθ is tngent to the ellipse t the point ( cos θ, sin θ). NOTE : (i) There re two tngents to the ellipse hving the sme m, i.e. there re two tngents prllel to n given direction.these tngents touches the ellipse t etremities of dimeter. (ii) Point of intersection of the tngents t the point α & β is, cos cos αβ α β, sin cos α β α β 7 of 9CONIC SECTION TEKO CLASSES, H.O.D. MATHS : SUHAG R. KARIYA (S. R. K. Sir) PH: (0755) , , BHOPAL, (M.P.)
8 FREE Downlod Stud Pckge from wesite: (iii) The eccentric ngles of the points of contct of two prllel tngents differ π. Solved Emple # 5 Find the equtions of the tngents to the ellipse 3 4 which re perpendiculr to the line 4. Slope of tngent m Given ellipse is 4 3 Eqution of tngent whose slope is 'm' is m ± 4m 3 m ± 3 ± 4 Solved Emple # 6 A tngent to the ellipse touches t the point P on it in the first qudrnt nd meets the co-ordinte es in A nd B respectivel. If P divides AB in the rtio 3 :, find the eqution of the tngent. Let P ( cosθ, sinθ) eqution of tngent is cosθ sinθ A ( secθ, 0) B (0, cosecθ) P divide AB internll in the rtio 3 : sec θ cosθ 4 nd sin θ 3cosecθ 4 cos θ 4 sinθ 3 cosθ 3 tngent is 3 Solved Emple # 7 Prove tht the locus of the point of intersection of tngents to n ellipse t two points whose eccentric ngle differ constnt α is n ellipse. Let P (h, k) e the point of intersection of tngents t A(θ) nd B(β) to the ellipse. h θ β cos θ β cos & k h k θ β sec ut given tht θ β α locus is Solved Emple # 8 α sec sec θ β sin θ β cos α Find the locus of foot of perpendiculr drwn from centre to n tngent to the ellipse is Let P(h, k) e the foot of perpendiculr to tngent m m...(i) from centre k h. m m h k P(h, k) lies on tngent...(ii). 8 of 9CONIC SECTION TEKO CLASSES, H.O.D. MATHS : SUHAG R. KARIYA (S. R. K. Sir) PH: (0755) , , BHOPAL, (M.P.)
9 FREE Downlod Stud Pckge from wesite: k mh m...(iii) from eqution (ii) & (iii), we get h k h k k locus is ( ) Self Prctice Prolem 4. Show tht the locus of the point of intersection of the tngents t the etremities of n focl chord of n ellipse is the directri corresponding to the focus. 5. Show tht the locus of the foot of the perpendiculr on vring tngent to n ellipse from either of its foci is concentric circle. 6. Prove tht the portion of the tngent to n ellipse intercepted etween the ellipse nd the directri sutends right ngle t the corresponding focus. 7. Find the re of prllelogrm formed tngents t the etremities of lter rect of the ellipse If is ordinte of point P on the ellipse then show tht the ngle etween its focl rdius nd tngent t it, is tn e. 9. Find the eccentric ngle of the point P on the ellipse inclined to the es. θ ± tn, π tn, π tn 8. N o r m l s : (i) Eqution of the norml t (, ) to the ellipse (ii) Eqution of the norml t the point (cos θ, sin θ) to the ellipse. sec θ. cosec θ (² ²). ( ) (iii) Eqution of norml in terms of its slope 'm' is m Solved Emple # 9 tngent t which, is equll is ² ². m. m is; 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 Sol. Let P (cos θ, sinθ) Q ( cosθ, sinθ) Eqution of norml t P is ( secθ) ( cosec θ)...(i) eqution of CQ is tnθ....(ii) Solving eqution (i) & (ii), we get ( ) ( ) cosθ ( ) cosθ, & ( ) sinθ R (( ) cosθ, ( ) sinθ) CR Solved Emple # 0 Find the shortest distnce etween the line 0 nd the ellipse of 9CONIC SECTION TEKO CLASSES, H.O.D. MATHS : SUHAG R. KARIYA (S. R. K. Sir) PH: (0755) , , BHOPAL, (M.P.)
10 FREE Downlod Stud Pckge from wesite: Shortest distnce occurs etween two non-intersecting curve lws long common norml. Let 'P' e point on ellipse nd Q is point on given line for which PQ is common norml. Tngent t 'P' is prllel to given line Eqution of tngent prllel to given line is ( m ± m ) ± or 5 0 minimum distnce distnce etween 0 0 & 5 0 shortest distnce Solved Emple # Prove tht, in n ellipse, the distnce etween the centre nd n norml does not eceed the difference etween the semi-es of the ellipse. Let the eqution of ellipse is Eqution of norml t P (θ) is ( secθ) (cosec θ) 0 distnce of norml from centre OR ( ) (tn θ) (tn θ cot θ) (cot θ) ( ) ( tnθ cotθ) ( ) or OR ( ) Self Prctice Prolem ( ) 0. Find the vlue(s) of 'k' for which the line k is norml to the ellipse ( ) k ±. If the norml t the point P(θ) to the ellipse cosθ (A*) 3 9. Pi r of Tngent s: (B) 3 intersects it gin t the point Q(θ) then 4 5 (C) 7 6 The eqution to the pir of tngents which cn e drwn from n point (, ) to the ellipse is given : SS T² where : S ; S ; T. Solved Emple # How mn rel tngents cn e drwn from the point (4, 3) to the ellipse 6 eqution these tngents & ngle etween them. Given point P (4, 3) 9 (D) 7 6. Find the 0 of 9CONIC SECTION TEKO CLASSES, H.O.D. MATHS : SUHAG R. KARIYA (S. R. K. Sir) PH: (0755) , , BHOPAL, (M.P.) ellipse S 6 0 9
11 FREE Downlod Stud Pckge from wesite: S > Point P (4, 3) lies outside the ellipse. Two tngents cn e drwn from the point P(4, 3). Eqution of pir of tngents is SS T (4 ) ( 3) 0 4 & 3 nd ngle etween them π Sol. E. # 3: Find the locus of point of intersection of perpendiculr tngents to the ellipse Let P(h, k) e the point of intersection of two perpendiculr tngents h k h k k h (i) Since eqution (i) represents two perpendiculr lines k h 0 k h 0 locus is Self Prctice Prolem :. Find the locus of point of intersection of the tngents drwn t the etremities of focl chord of the ellipse 0. Director Circle:. ± e 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 ² ² ² ² 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 es. Solved Emple # 4 An ellipse slides etween two perpendiculr lines. Show tht the locus of its centre is circle. Solution : Let length of semi-mjor nd semi-minor is re '' nd '' nd centre is C (h, k) Since ellipse slides etween two perpendiculr lines, there for point of intersection of two perpendiculr tngents lies on director circle. Let us consider two perpendiculr lines s & es point of intersection is origin O (0, 0) OC rdius of director circle h k locus of C (h, k) is which is circle Self Prctice Prolem A tngent to the ellipse 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.. Chord of Contct: Eqution to the chord of contct of tngents drwn from point P(, ) to the ellipse T 0, where T Solved Emple # 5 If tngents to the prol 4 intersect the ellipse of point of intersection of tngents t A nd B. is t A nd B, then find the locus of 9CONIC SECTION TEKO CLASSES, H.O.D. MATHS : SUHAG R. KARIYA (S. R. K. Sir) PH: (0755) , , BHOPAL, (M.P.)
12 FREE Downlod Stud Pckge from wesite: Solution: 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 eqution of tngent to prol 4 m m m m eqution (i) & (ii) s must e sme m h k m k h h k k m k & m 4...(ii) locus of P is 3. Self Prctice Prolem 3. Find the locus of point of intersection of tngents t the etremities of norml chords of the 6 ellipse. ( ) 4. Find the locus of point of intersection of tngents t the etremities of the chords of the ellipse 4 4 sutending right ngle t its centre.. Chord with given middle point: Eqution of the chord of the ellipse whose middle point is (, ) is T S, where S ; T. Solved Emple # 6 Find the locus of the mid - point of focl chords of the ellipse Solution: Let P (h, k) e the mid-point 6 h k h 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 trhrough (e, 0) e locus is If it psses through ( e, 0) e locus is Solved Emple # 7:Find the condition on '' nd '' for which two distinct chords of the ellipse pssing through (, ) re isected the line. Solution: Let the line isect the chord t P(α, α) eqution of chord whose mid-point is P(α, α) α ( α) α ( α). k of 9CONIC SECTION TEKO CLASSES, H.O.D. MATHS : SUHAG R. KARIYA (S. R. K. Sir) PH: (0755) , , BHOPAL, (M.P.)
13 Since it psses through (, ) α ( α) α ( α) α α α α since line isect two chord ove qudrtic eqution in α must hve two distinct rel roots 3 4. > 0 3 α 0 3 of 9CONIC SECTION FREE Downlod Stud Pckge from wesite: > 0 6 > > 0 > 7 6 which is the required condition. Self Prctice Prolem 5. Find the eqution of the chord which is isected t (, ) Find the locus of the mid-points of norml chords of the ellipse 6 6 ( ) 7. Find the length of the chord of the ellipse whose middle point is, Importnt High Lights : Refering to the ellipse If P e n point on the ellipse with S & S s its foci then l (SP) l (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. Hence we cn deduce tht the stright lines joining ech focus to the foot of the perpendiculr from the other focus upon the tngent t n point P meet on the norml PG nd isects it where G is the point where norml t P meets the mjor is. The product of the length s of the perpendiculr segments from the foci on n tngent to the ellipse is ² 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 similir ellipse s tht of the originl one. The portion of the tngent to n ellipse etween the point of contct & the directri sutends right ngle t the corresponding focus. 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 ² (ii) PF. Pg ² (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] The circle on n focl distnce s dimeter touches the uilir circle. Perpendiculrs from the centre upon ll chords which join the ends of n perpendiculr dimeters of the ellipse re of constnt length. If the tngent t the point P of stndrd ellipse meets the is in T nd t nd CY is the perpendiculr on it from the centre then, (i) T t. PY nd (ii) lest vlue of T t is.. TEKO CLASSES, H.O.D. MATHS : SUHAG R. KARIYA (S. R. K. Sir) PH: (0755) , , BHOPAL, (M.P.)
Lesson-5 ELLIPSE 2 1 = 0
Lesson-5 ELLIPSE. An ellipse is the locus of point which moves in plne such tht its distnce from fied point (known s the focus) is e (< ), times its distnce from fied stright line (known s the directri).
More information, MATHS H.O.D.: SUHAG R.KARIYA, BHOPAL, CONIC SECTION PART 8 OF
DOWNLOAD FREE FROM www.tekoclsses.com, PH.: 0 903 903 7779, 98930 5888 Some questions (Assertion Reson tpe) re given elow. Ech question contins Sttement (Assertion) nd Sttement (Reson). Ech question hs
More informationJEE Advnced Mths Assignment Onl One Correct Answer Tpe. The locus of the orthocenter of the tringle formed the lines (+P) P + P(+P) = 0, (+q) q+q(+q) = 0 nd = 0, where p q, is () hperol prol n ellipse
More information/ 3, then (A) 3(a 2 m 2 + b 2 ) = 4c 2 (B) 3(a 2 + b 2 m 2 ) = 4c 2 (C) a 2 m 2 + b 2 = 4c 2 (D) a 2 + b 2 m 2 = 4c 2
SET I. If the locus of the point of intersection of perpendiculr tngents to the ellipse x circle with centre t (0, 0), then the rdius of the circle would e + / ( ) is. There re exctl two points on the
More informationCONIC SECTIONS. Chapter 11
CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round
More informationDrill Exercise Find the coordinates of the vertices, foci, eccentricity and the equations of the directrix of the hyperbola 4x 2 25y 2 = 100.
Drill Exercise - 1 1 Find the coordintes of the vertices, foci, eccentricit nd the equtions of the directrix of the hperol 4x 5 = 100 Find the eccentricit of the hperol whose ltus-rectum is 8 nd conjugte
More informationELLIPSE. Standard equation of an ellipse referred to its principal axes along the co-ordinate axes is. ( a,0) A'
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
More informationNORMALS. a y a y. Therefore, the slope of the normal is. a y1. b x1. b x. a b. x y a b. x y
LOCUS 50 Section - 4 NORMALS Consider n ellipse. We need to find the eqution of the norml to this ellipse t given point P on it. In generl, we lso need to find wht condition must e stisfied if m c is to
More informationHYPERBOLA. AIEEE Syllabus. Total No. of questions in Ellipse are: Solved examples Level # Level # Level # 3..
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
More informationP 1 (x 1, y 1 ) is given by,.
MA00 Clculus nd Bsic Liner Alger I Chpter Coordinte Geometr nd Conic Sections Review In the rectngulr/crtesin coordintes sstem, we descrie the loction of points using coordintes. P (, ) P(, ) O The distnce
More informationPARABOLA EXERCISE 3(B)
PARABOLA EXERCISE (B). Find eqution of the tngent nd norml to the prbol y = 6x t the positive end of the ltus rectum. Eqution of prbol y = 6x 4 = 6 = / Positive end of the Ltus rectum is(, ) =, Eqution
More informationMATHEMATICS PAPER IIB COORDINATE GEOMETRY AND CALCULUS. Note: This question paper consists of three sections A,B and C. SECTION A
MATHEMATICS PAPER IIB COORDINATE GEOMETRY AND CALCULUS. TIME : 3hrs M. Mrks.75 Note: This question pper consists of three sections A,B nd C. SECTION A VERY SHORT ANSWER TYPE QUESTIONS. X = ) Find the eqution
More information1. If y 2 2x 2y + 5 = 0 is (A) a circle with centre (1, 1) (B) a parabola with vertex (1, 2) 9 (A) 0, (B) 4, (C) (4, 4) (D) a (C) c = am m.
SET I. If y x y + 5 = 0 is (A) circle with centre (, ) (B) prbol with vertex (, ) (C) prbol with directrix x = 3. The focus of the prbol x 8x + y + 7 = 0 is (D) prbol with directrix x = 9 9 (A) 0, (B)
More informationTime : 3 hours 03 - Mathematics - March 2007 Marks : 100 Pg - 1 S E CT I O N - A
Time : hours 0 - Mthemtics - Mrch 007 Mrks : 100 Pg - 1 Instructions : 1. Answer ll questions.. Write your nswers ccording to the instructions given below with the questions.. Begin ech section on new
More informationMathematics. Area under Curve.
Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding
More informationR(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of
Higher Mthemtics Ojective Test Prctice ook The digrm shows sketch of prt of the grph of f ( ). The digrm shows sketch of the cuic f ( ). R(, 8) f ( ) f ( ) P(, ) Q(, ) S(, ) Wht re the domin nd rnge of
More informationCET MATHEMATICS 2013
CET MATHEMATICS VERSION CODE: C. If sin is the cute ngle between the curves + nd + 8 t (, ), then () () () Ans: () Slope of first curve m ; slope of second curve m - therefore ngle is o A sin o (). The
More information1. If * is the operation defined by a*b = a b for a, b N, then (2 * 3) * 2 is equal to (A) 81 (B) 512 (C) 216 (D) 64 (E) 243 ANSWER : D
. If * is the opertion defined by *b = b for, b N, then ( * ) * is equl to (A) 8 (B) 5 (C) 6 (D) 64 (E) 4. The domin of the function ( 9)/( ),if f( ) = is 6, if = (A) (0, ) (B) (-, ) (C) (-, ) (D) (, )
More informationLinear Inequalities: Each of the following carries five marks each: 1. Solve the system of equations graphically.
Liner Inequlities: Ech of the following crries five mrks ech:. Solve the system of equtions grphiclly. x + 2y 8, 2x + y 8, x 0, y 0 Solution: Considerx + 2y 8.. () Drw the grph for x + 2y = 8 by line.it
More informationI. Equations of a Circle a. At the origin center= r= b. Standard from: center= r=
11.: Circle & Ellipse I cn Write the eqution of circle given specific informtion Grph circle in coordinte plne. Grph n ellipse nd determine ll criticl informtion. Write the eqution of n ellipse from rel
More informationk ) and directrix x = h p is A focal chord is a line segment which passes through the focus of a parabola and has endpoints on the parabola.
Stndrd Eqution of Prol with vertex ( h, k ) nd directrix y = k p is ( x h) p ( y k ) = 4. Verticl xis of symmetry Stndrd Eqution of Prol with vertex ( h, k ) nd directrix x = h p is ( y k ) p( x h) = 4.
More informationMH CET 2018 (QUESTION WITH ANSWER)
( P C M ) MH CET 8 (QUESTION WITH ANSWER). /.sec () + log () - log (3) + log () Ans. () - log MATHS () 3 c + c C C A cos + cos c + cosc + + cosa ( + cosc ) + + cosa c c ( + + ) c / / I tn - in sec - in
More informationKEY CONCEPTS. satisfies the differential equation da. = 0. Note : If F (x) is any integral of f (x) then, x a
KEY CONCEPTS THINGS TO REMEMBER :. The re ounded y the curve y = f(), the -is nd the ordintes t = & = is given y, A = f () d = y d.. If the re is elow the is then A is negtive. The convention is to consider
More informationPrerequisite Knowledge Required from O Level Add Math. d n a = c and b = d
Prerequisite Knowledge Required from O Level Add Mth ) Surds, Indices & Logrithms Rules for Surds. b= b =. 3. 4. b = b = ( ) = = = 5. + b n = c+ d n = c nd b = d Cution: + +, - Rtionlising the Denomintor
More information( β ) touches the x-axis if = 1
Generl Certificte of Eduction (dv. Level) Emintion, ugust Comined Mthemtics I - Prt B Model nswers. () Let f k k, where k is rel constnt. i. Epress f in the form( ) Find the turning point of f without
More informationSCORE JEE (Advanced)
SLUTIN. ns. (D) L : x + y 0 S L : x + y 0 L : x + y 7 0 Point of intersection of L 0 & L 0 is (,9) Point of intersection of L 0 & L 0 is (0,) line perpendiculr to L nd pssing through (, 9) isx y + 0...
More informationPROPERTIES OF AREAS In general, and for an irregular shape, the definition of the centroid at position ( x, y) is given by
PROPERTES OF RES Centroid The concept of the centroid is prol lred fmilir to ou For plne shpe with n ovious geometric centre, (rectngle, circle) the centroid is t the centre f n re hs n is of smmetr, the
More informationSketch graphs of conic sections and write equations related to conic sections
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
More informationFP3 past questions - conics
Hperolic functions cosh sinh = sinh = sinh cosh cosh = cosh + sinh rcosh = ln{ + } ( ) rsinh = ln{ + + } + rtnh = ln ( < ) FP3 pst questions - conics Conics Ellipse Prol Hperol Rectngulr Hperol Stndrd
More informationEigen Values and Eigen Vectors of a given matrix
Engineering Mthemtics 0 SUBJECT NAME SUBJECT CODE MATERIAL NAME MATERIAL CODE : Engineering Mthemtics I : 80/MA : Prolem Mteril : JM08AM00 (Scn the ove QR code for the direct downlod of this mteril) Nme
More informationMATHEMATICS PART A. 1. ABC is a triangle, right angled at A. The resultant of the forces acting along AB, AC
FIITJEE Solutions to AIEEE MATHEMATICS PART A. ABC is tringle, right ngled t A. The resultnt of the forces cting long AB, AC with mgnitudes AB nd respectively is the force long AD, where D is the AC foot
More informationAlgebra II Notes Unit Ten: Conic Sections
Syllus Ojective: 10.1 The student will sketch the grph of conic section with centers either t or not t the origin. (PARABOLAS) Review: The Midpoint Formul The midpoint M of the line segment connecting
More information1 CONIC SECTIONS While cutting crrot ou might hve noticed different shpes shown b the edges of the cut. Anlticll ou m cut it in three different ws, nmel (i) (ii) (iii) Cut is prllel to the bse (see Fig.1.1)
More informationParabola Exercise 1 2,6 Q.1 (A) S(0, 1) directric x + 2y = 0 PS = PM. x y x y 2y 1 x 2y Q.2 (D) y 2 = 18 x. 2 = 3t. 2 t 3 Q.
Prbol Exercise Q. (A) S(0, ) directric x + y = 0 PS = PM x y x y 5 5 x y y x y Q. (D) y = 8 x (t, t) t t = t t 8 4 8 t,t, 4 9 4,6 Q. (C) y 4 x 5 Eqution of directrix is x + = 0 x 0 5 Q.4 y = 8x M P t,t
More informationCoimisiún na Scrúduithe Stáit State Examinations Commission
M 30 Coimisiún n Scrúduithe Stáit Stte Exmintions Commission LEAVING CERTIFICATE EXAMINATION, 005 MATHEMATICS HIGHER LEVEL PAPER ( 300 mrks ) MONDAY, 3 JUNE MORNING, 9:30 to :00 Attempt FIVE questions
More informationJEE(MAIN) 2015 TEST PAPER WITH SOLUTION (HELD ON SATURDAY 04 th APRIL, 2015) PART B MATHEMATICS
JEE(MAIN) 05 TEST PAPER WITH SOLUTION (HELD ON SATURDAY 0 th APRIL, 05) PART B MATHEMATICS CODE-D. Let, b nd c be three non-zero vectors such tht no two of them re colliner nd, b c b c. If is the ngle
More informationm m m m m m m m P m P m ( ) m m P( ) ( ). The o-ordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r
CO-ORDINTE GEOMETR II I Qudrnt Qudrnt (-.+) (++) X X - - - 0 - III IV Qudrnt - Qudrnt (--) - (+-) Region CRTESIN CO-ORDINTE SSTEM : Retngulr Co-ordinte Sstem : Let X' OX nd 'O e two mutull perpendiulr
More informationThomas Whitham Sixth Form
Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos
More informationLoudoun Valley High School Calculus Summertime Fun Packet
Loudoun Vlley High School Clculus Summertime Fun Pcket We HIGHLY recommend tht you go through this pcket nd mke sure tht you know how to do everything in it. Prctice the problems tht you do NOT remember!
More informationMATHEMATICS (Part II) (Fresh / New Course)
Sig. of Supdt... MRD-XII-(A) MATHEMATICS Roll No... Time Allowed : Hrs. MATHEMATICS Totl Mrks: 00 NOTE : There re THREE sections in this pper i.e. Section A, B nd C. Time : 0 Mins. Section A Mrks: 0 NOTE
More information1 is equal to. 1 (B) a. (C) a (B) (D) 4. (C) P lies inside both C & E (D) P lies inside C but outside E. (B) 1 (D) 1
Single Correct Q. Two mutuall perpendicular tangents of the parabola = a meet the ais in P and P. If S is the focus of the parabola then l a (SP ) is equal to (SP ) l (B) a (C) a Q. ABCD and EFGC are squares
More information15 - TRIGONOMETRY Page 1 ( Answers at the end of all questions )
- TRIGONOMETRY Pge P ( ) In tringle PQR, R =. If tn b c = 0, 0, then Q nd tn re the roots of the eqution = b c c = b b = c b = c [ AIEEE 00 ] ( ) In tringle ABC, let C =. If r is the inrdius nd R is the
More informationby Abhijit Kumar Jha
SET I. If the locus of the point of intersection of perpendicular tangents to the ellipse x a circle with centre at (0, 0), then the radius of the circle would e a + a /a ( a ). There are exactl two points
More informationJUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 12 (Second moments of an area (B)) A.J.Hobson
JUST THE MATHS UNIT NUMBE 13.1 INTEGATION APPLICATIONS 1 (Second moments of n re (B)) b A.J.Hobson 13.1.1 The prllel xis theorem 13.1. The perpendiculr xis theorem 13.1.3 The rdius of grtion of n re 13.1.4
More informationJUST THE MATHS SLIDES NUMBER INTEGRATION APPLICATIONS 12 (Second moments of an area (B)) A.J.Hobson
JUST THE MATHS SLIDES NUMBER 13.12 INTEGRATION APPLICATIONS 12 (Second moments of n re (B)) b A.J.Hobson 13.12.1 The prllel xis theorem 13.12.2 The perpendiculr xis theorem 13.12.3 The rdius of grtion
More informationPolynomials and Division Theory
Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the
More informationSECTION 9-4 Translation of Axes
9-4 Trnsltion of Aes 639 Rdiotelescope For the receiving ntenn shown in the figure, the common focus F is locted 120 feet bove the verte of the prbol, nd focus F (for the hperbol) is 20 feet bove the verte.
More informationSUBJECT: MATHEMATICS ANSWERS: COMMON ENTRANCE TEST 2012
MOCK TEST 0 SUBJECT: MATHEMATICS ANSWERS: COMMON ENTRANCE TEST 0 ANSWERS. () π π Tke cos - (- ) then sin [ cos - (- )]sin [ ]/. () Since sin - + sin - y + sin - z π, -; y -, z - 50 + y 50 + z 50 - + +
More informationMathematics Extension 2
00 HIGHER SCHOOL CERTIFICATE EXAMINATION Mthemtics Extension Generl Instructions Reding time 5 minutes Working time hours Write using blck or blue pen Bord-pproved clcultors m be used A tble of stndrd
More informationEdexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks
Edexcel GCE Core Mthemtics (C) Required Knowledge Informtion Sheet C Formule Given in Mthemticl Formule nd Sttisticl Tles Booklet Cosine Rule o = + c c cosine (A) Binomil Series o ( + ) n = n + n 1 n 1
More informationMathematics Extension 2
00 HIGHER SCHOOL CERTIFICATE EXAMINATION Mthemtics Etension Generl Instructions Reding time 5 minutes Working time hours Write using blck or blue pen Bord-pproved clcultors my be used A tble of stndrd
More informationS56 (5.3) Vectors.notebook January 29, 2016
Dily Prctice 15.1.16 Q1. The roots of the eqution (x 1)(x + k) = 4 re equl. Find the vlues of k. Q2. Find the rte of chnge of 剹 x when x = 1 / 8 Tody we will e lerning out vectors. Q3. Find the eqution
More informationMATHEMATICS IV 2 MARKS. 5 2 = e 3, 4
MATHEMATICS IV MARKS. If + + 6 + c epesents cicle with dius 6, find the vlue of c. R 9 f c ; g, f 6 9 c 6 c c. Find the eccenticit of the hpeol Eqution of the hpeol Hee, nd + e + e 5 e 5 e. Find the distnce
More informationGEOMETRY OF THE CIRCLE TANGENTS & SECANTS
Geometry Of The ircle Tngents & Secnts GEOMETRY OF THE IRLE TNGENTS & SENTS www.mthletics.com.u Tngents TNGENTS nd N Secnts SENTS Tngents nd secnts re lines tht strt outside circle. Tngent touches the
More informationTriangles The following examples explore aspects of triangles:
Tringles The following exmples explore spects of tringles: xmple 1: ltitude of right ngled tringle + xmple : tringle ltitude of the symmetricl ltitude of n isosceles x x - 4 +x xmple 3: ltitude of the
More informationOn the diagram below the displacement is represented by the directed line segment OA.
Vectors Sclrs nd Vectors A vector is quntity tht hs mgnitude nd direction. One exmple of vector is velocity. The velocity of n oject is determined y the mgnitude(speed) nd direction of trvel. Other exmples
More informationUS01CMTH02 UNIT Curvature
Stu mteril of BSc(Semester - I) US1CMTH (Rdius of Curvture nd Rectifiction) Prepred by Nilesh Y Ptel Hed,Mthemtics Deprtment,VPnd RPTPScience College US1CMTH UNIT- 1 Curvture Let f : I R be sufficiently
More information+ R 2 where R 1. MULTIPLE CHOICE QUESTIONS (MCQ's) (Each question carries one mark)
2. C h p t e r t G l n c e is the set of ll points in plne which re t constnt distnce from fixed point clled centre nd constnt distnce is known s rdius of circle. A tngent t ny point of circle is perpendiculr
More informationAlg. Sheet (1) Department : Math Form : 3 rd prep. Sheet
Ciro Governorte Nozh Directorte of Eduction Nozh Lnguge Schools Ismili Rod Deprtment : Mth Form : rd prep. Sheet Alg. Sheet () [] Find the vlues of nd in ech of the following if : ) (, ) ( -5, 9 ) ) (,
More informationPrerna Tower, Road No 2, Contractors Area, Bistupur, Jamshedpur , Tel (0657) ,
R rern Tower, Rod No, Contrctors Are, Bistupur, Jmshedpur 800, Tel 065789, www.prernclsses.com IIT JEE 0 Mthemtics per I ART III SECTION I Single Correct Answer Type This section contins 0 multiple choice
More informationJUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 6 (First moments of an arc) A.J.Hobson
JUST THE MATHS UNIT NUMBER 13.6 INTEGRATION APPLICATIONS 6 (First moments of n rc) by A.J.Hobson 13.6.1 Introduction 13.6. First moment of n rc bout the y-xis 13.6.3 First moment of n rc bout the x-xis
More informationBRIEF NOTES ADDITIONAL MATHEMATICS FORM
BRIEF NOTES ADDITIONAL MATHEMATICS FORM CHAPTER : FUNCTION. : + is the object, + is the imge : + cn be written s () = +. To ind the imge or mens () = + = Imge or is. Find the object or 8 mens () = 8 wht
More informationIMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHS-IB
` K UKATP ALLY CE NTRE IMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHS-IB 7-8 FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Ptel Kunt Hud Prk, Vijngr Colon, Hderbd - 5 7 Ph: -66 Regd
More informationA quick overview of the four conic sections in rectangular coordinates is presented below.
MAT 6H Rectngulr Equtions of Conics A quick overview of the four conic sections in rectngulr coordintes is presented elow.. Circles Skipped covered in previous lger course.. Prols Definition A prol is
More information03 Qudrtic Functions Completing the squre: Generl Form f ( x) x + x + c f ( x) ( x + p) + q where,, nd c re constnts nd 0. (i) (ii) (iii) (iv) *Note t
A-PDF Wtermrk DEMO: Purchse from www.a-pdf.com to remove the wtermrk Add Mths Formule List: Form 4 (Updte 8/9/08) 0 Functions Asolute Vlue Function Inverse Function If f ( x ), if f ( x ) 0 f ( x) y f
More informationat its center, then the measure of this angle in radians (abbreviated rad) is the length of the arc that subtends the angle.
Notes 6 ngle Mesure Definition of Rdin If circle of rdius is drwn with the vertex of n ngle Mesure: t its center, then the mesure of this ngle in rdins (revited rd) is the length of the rc tht sutends
More information8.6 The Hyperbola. and F 2. is a constant. P F 2. P =k The two fixed points, F 1. , are called the foci of the hyperbola. The line segments F 1
8. The Hperol Some ships nvigte using rdio nvigtion sstem clled LORAN, which is n cronm for LOng RAnge Nvigtion. A ship receives rdio signls from pirs of trnsmitting sttions tht send signls t the sme time.
More informationEdexcel GCE A Level Maths. Further Maths 3 Coordinate Systems
Edecel GCE A Level Maths Further Maths 3 Coordinate Sstems Edited b: K V Kumaran kumarmaths.weebl.com 1 kumarmaths.weebl.com kumarmaths.weebl.com 3 kumarmaths.weebl.com 4 kumarmaths.weebl.com 5 1. An ellipse
More informationForm 5 HKCEE 1990 Mathematics II (a 2n ) 3 = A. f(1) B. f(n) A. a 6n B. a 8n C. D. E. 2 D. 1 E. n. 1 in. If 2 = 10 p, 3 = 10 q, express log 6
Form HK 9 Mthemtics II.. ( n ) =. 6n. 8n. n 6n 8n... +. 6.. f(). f(n). n n If = 0 p, = 0 q, epress log 6 in terms of p nd q.. p q. pq. p q pq p + q Let > b > 0. If nd b re respectivel the st nd nd terms
More informationBoard Answer Paper: October 2014
Trget Pulictions Pvt. Ltd. Bord Answer Pper: Octoer 4 Mthemtics nd Sttistics SECTION I Q.. (A) Select nd write the correct nswer from the given lterntives in ech of the following su-questions: i. (D) ii..p
More informationFORM FIVE ADDITIONAL MATHEMATIC NOTE. ar 3 = (1) ar 5 = = (2) (2) (1) a = T 8 = 81
FORM FIVE ADDITIONAL MATHEMATIC NOTE CHAPTER : PROGRESSION Arithmetic Progression T n = + (n ) d S n = n [ + (n )d] = n [ + Tn ] S = T = T = S S Emple : The th term of n A.P. is 86 nd the sum of the first
More informationDE51/DC51 ENGINEERING MATHEMATICS I DEC 2013
DE5/DC5 ENGINEERING MATHEMATICS I DEC π π Q.. Prove tht cos α + cos α + + cos α + L.H.S. π π cos α + cos α + + cos α + + α + cos α + cos ( α ) + cos ( ) cos α + cos ( 9 + α + ) + cos(8 + α + 6 ) cos α
More informationSTRAND J: TRANSFORMATIONS, VECTORS and MATRICES
Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors
More informationPARABOLA. moves such that PM. = e (constant > 0) (eccentricity) then locus of P is called a conic. or conic section.
wwwskshieducioncom PARABOLA Le S be given fixed poin (focus) nd le l be given fixed line (Direcrix) Le SP nd PM be he disnce of vrible poin P o he focus nd direcrix respecively nd P SP moves such h PM
More informationELLIPSE. 1. If the latus rectum of an ellipse be equal to half of its minor axis, then its eccentricity is [Karnataka CET 2000]
ELLIPSE. If the ltus rectum of ellipse e equl to hlf of its mior is, the its eccetricit is [Krtk CET 000] / / / d /. The legth of the ltus rectum of the ellipse is [MNR 7, 0, ] / / / d 0/. Eccetricit of
More informationES.182A Topic 32 Notes Jeremy Orloff
ES.8A Topic 3 Notes Jerem Orloff 3 Polr coordintes nd double integrls 3. Polr Coordintes (, ) = (r cos(θ), r sin(θ)) r θ Stndrd,, r, θ tringle Polr coordintes re just stndrd trigonometric reltions. In
More informationMEP Practice Book ES3. 1. Calculate the size of the angles marked with a letter in each diagram. None to scale
ME rctice ook ES3 3 ngle Geometr 3.3 ngle Geometr 1. lculte the size of the ngles mrked with letter in ech digrm. None to scle () 70 () 20 54 65 25 c 36 (d) (e) (f) 56 62 d e 60 40 70 70 f 30 g (g) (h)
More informationObjective Mathematics
. A tangent to the ellipse is intersected by a b the tangents at the etremities of the major ais at 'P' and 'Q' circle on PQ as diameter always passes through : (a) one fied point two fied points (c) four
More informationMath 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 2 solutions
Mth 1102: Clculus I (Mth/Sci mjors) MWF 3pm, Fulton Hll 230 Homework 2 solutions Plese write netly, nd show ll work. Cution: An nswer with no work is wrong! Do the following problems from Chpter III: 6,
More informationUse of Trigonometric Functions
Unit 03 Use of Trigonometric Functions 1. Introduction Lerning Ojectives of tis UNIT 1. Lern ow te trigonometric functions re relted to te rtios of sides of rigt ngle tringle. 2. Be le to determine te
More information10.5. ; 43. The points of intersection of the cardioid r 1 sin and. ; Graph the curve and find its length. CONIC SECTIONS
654 CHAPTER 1 PARAETRIC EQUATIONS AND POLAR COORDINATES ; 43. The points of intersection of the crdioid r 1 sin nd the spirl loop r,, cn t be found ectl. Use grphing device to find the pproimte vlues of
More informationTime : 3 hours 02 - Mathematics - July 2006 Marks : 100 Pg - 1 Instructions : S E CT I O N - A
Time : 3 hours 0 Mathematics July 006 Marks : 00 Pg Instructions :. Answer all questions.. Write your answers according to the instructions given below with the questions. 3. Begin each section on a new
More informationOptimization Lecture 1 Review of Differential Calculus for Functions of Single Variable.
Optimiztion Lecture 1 Review of Differentil Clculus for Functions of Single Vrible http://users.encs.concordi.c/~luisrod, Jnury 14 Outline Optimiztion Problems Rel Numbers nd Rel Vectors Open, Closed nd
More informationMathematics Extension Two
Student Number 04 HSC TRIAL EXAMINATION Mthemtics Etension Two Generl Instructions Reding time 5 minutes Working time - hours Write using blck or blue pen Bord-pproved clcultors my be used Write your Student
More informationH (2a, a) (u 2a) 2 (E) Show that u v 4a. Explain why this implies that u v 4a, with equality if and only u a if u v 2a.
Chpter Review 89 IGURE ol hord GH of the prol 4. G u v H (, ) (A) Use the distne formul to show tht u. (B) Show tht G nd H lie on the line m, where m ( )/( ). (C) Solve m for nd sustitute in 4, otining
More information( x )( x) dx. Year 12 Extension 2 Term Question 1 (15 Marks) (a) Sketch the curve (x + 1)(y 2) = 1 2
Yer Etension Term 7 Question (5 Mrks) Mrks () Sketch the curve ( + )(y ) (b) Write the function in prt () in the form y f(). Hence, or otherwise, sketch the curve (i) y f( ) (ii) y f () (c) Evlute (i)
More informationYear 12 Mathematics Extension 2 HSC Trial Examination 2014
Yer Mthemtics Etension HSC Tril Emintion 04 Generl Instructions. Reding time 5 minutes Working time hours Write using blck or blue pen. Blck pen is preferred. Bord-pproved clcultors my be used A tble of
More informationGEOMETRICAL PROPERTIES OF ANGLES AND CIRCLES, ANGLES PROPERTIES OF TRIANGLES, QUADRILATERALS AND POLYGONS:
GEOMETRICL PROPERTIES OF NGLES ND CIRCLES, NGLES PROPERTIES OF TRINGLES, QUDRILTERLS ND POLYGONS: 1.1 TYPES OF NGLES: CUTE NGLE RIGHT NGLE OTUSE NGLE STRIGHT NGLE REFLEX NGLE 40 0 4 0 90 0 156 0 180 0
More informationSection 13.1 Right Triangles
Section 13.1 Right Tringles Ojectives: 1. To find vlues of trigonometric functions for cute ngles. 2. To solve tringles involving right ngles. Review - - 1. SOH sin = Reciprocl csc = 2. H cos = Reciprocl
More informationA LEVEL TOPIC REVIEW. factor and remainder theorems
A LEVEL TOPIC REVIEW unit C fctor nd reminder theorems. Use the Fctor Theorem to show tht: ) ( ) is fctor of +. ( mrks) ( + ) is fctor of ( ) is fctor of + 7+. ( mrks) +. ( mrks). Use lgebric division
More information2. VECTORS AND MATRICES IN 3 DIMENSIONS
2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the
More informationIntegration Techniques
Integrtion Techniques. Integrtion of Trigonometric Functions Exmple. Evlute cos x. Recll tht cos x = cos x. Hence, cos x Exmple. Evlute = ( + cos x) = (x + sin x) + C = x + 4 sin x + C. cos 3 x. Let u
More informationMTH 4-16a Trigonometry
MTH 4-16 Trigonometry Level 4 [UNIT 5 REVISION SECTION ] I cn identify the opposite, djcent nd hypotenuse sides on right-ngled tringle. Identify the opposite, djcent nd hypotenuse in the following right-ngled
More informationMinnesota State University, Mankato 44 th Annual High School Mathematics Contest April 12, 2017
Minnesot Stte University, Mnkto 44 th Annul High School Mthemtics Contest April, 07. A 5 ft. ldder is plced ginst verticl wll of uilding. The foot of the ldder rests on the floor nd is 7 ft. from the wll.
More informationC Precalculus Review. C.1 Real Numbers and the Real Number Line. Real Numbers and the Real Number Line
C. Rel Numers nd the Rel Numer Line C C Preclculus Review C. Rel Numers nd the Rel Numer Line Represent nd clssif rel numers. Order rel numers nd use inequlities. Find the solute vlues of rel numers nd
More informationAnswers: ( HKMO Heat Events) Created by: Mr. Francis Hung Last updated: 15 December 2017
Answers: (0- HKMO Het Events) reted y: Mr. Frncis Hung Lst updted: 5 Decemer 07 - Individul - Group Individul Events 6 80 0 4 5 5 0 6 4 7 8 5 9 9 0 9 609 4 808 5 0 6 6 7 6 8 0 9 67 0 0 I Simplify 94 0.
More informationCBSE-XII-2015 EXAMINATION. Section A. 1. Find the sum of the order and the degree of the following differential equation : = 0
CBSE-XII- EXMINTION MTHEMTICS Pper & Solution Time : Hrs. M. Mrks : Generl Instruction : (i) ll questions re compulsory. There re questions in ll. (ii) This question pper hs three sections : Section, Section
More informationPRACTICE PAPER 6 SOLUTIONS
PRACTICE PAPER 6 SOLUTIONS SECTION A I.. Find the value of k if the points (, ) and (k, 3) are conjugate points with respect to the circle + y 5 + 8y + 6. Sol. Equation of the circle is + y 5 + 8y + 6
More informationalong the vector 5 a) Find the plane s coordinate after 1 hour. b) Find the plane s coordinate after 2 hours. c) Find the plane s coordinate
L8 VECTOR EQUATIONS OF LINES HL Mth - Sntowski Vector eqution of line 1 A plne strts journey t the point (4,1) moves ech hour long the vector. ) Find the plne s coordinte fter 1 hour. b) Find the plne
More informationSet 6 Paper 2. Set 6 Paper 2. 1 Pearson Education Asia Limited 2017
Set 6 Pper Set 6 Pper. C. C. A. D. B 6. D 7. D 8. A 9. D 0. A. B. B. A. B. B 6. B 7. D 8. C 9. D 0. D. A. A. B. B. C 6. C 7. A 8. B 9. A 0. A. C. D. B. B. B 6. A 7. D 8. A 9. C 0. C. C. D. C. C. D Section
More information