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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 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).