Lesson-5 ELLIPSE 2 1 = 0

Transcription

1 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). The constnt e is clled s the eccentricit of the ellipse.. Eqution of ellipse in stndrd form, where > nd ( e ). () Center is C(0, 0) () S(e, 0) nd S( e, 0) re the two foci (c) z z nd z z re the two directrices corresponding to the foci S nd S respectivel. z A L S B C S L P A z M z z hs eqution hs eqution e e while z z z L B L z (d) A(, 0) nd A(, 0) re the two vertices. (e) The line AA joining foci is the mjor is nd hs eqution 0; its length is. (f) The line joining B(0, ) nd B(0, ) is the minor is nd hs eqution: 0; its length is. (g) The lines L SL nd L SL re the two lter rect nd ech hs length.. Alternte definition: An ellipse is locus of point P which moves in plne such tht the sum of its distnces from the two foci S nd S is constnt. Thus PS + PS AA (length of mjor is) where > SS.. The generl eqution of the second degree h g S + h + h f + g + f + c 0 represents n ellipse if 0 nd h <. g f c Corresponding to point P(, ), we define, for the ellipse S 0, S + h + + g + f + c T + h( + ) + + g( + ) + f ( + ) + c 5. Position of point w.r.t the ellipse 0 The point P(, ) lies within, on or outside S 0 ccording s S < 0, 0 or > 0.

2 6. Auillr circle The circle with center sme s center of the ellipse nd dimeter hving length of mjor is is clled s uillr circle. For the ellipse, where >, the eqution of uillr circle is +. -is If P lies on the ellipse, the line PN perpendiculr to mjor is cuts uillr circle t P Q( cos, sin ) where 0 <. Thus, P hs coordintes ( cos, sin ) nd is clled s eccentric ngle of P. N -is 7. Tngents nd norml Eqution of tngent to S 0 t P(, ) is T 0. Eqution of tngent Q () t P(, ) to is: () t P( cos, sin ) is: cos sin (c) in slope form is: m ± m, where m is the slope of tngent. Eqution of norml to () t P(, ) is () t P( cos, sin ) is: sec cosec 8. Pir of tngents: The joint eqution of the pir of tngents to S 0 from n eternl point P(, ) is SS T.

3 9. Chord of contct The eqution of the chord of contct of tngents drwn to S 0 from n eternl point P(, ) is T 0 0. Eqution of chord with middle point (, ) is T S. Director circle It is the locus of the point of intersection of tngents which re perpendiculr to ech other. Its eqution, with respect to the ellipse, is + +. Eqution of chord joining the points ( cos, sin ) nd ( cos, sin ) cos sin cos Ellipse with mjor es long -es. For the ellipse, if <, then the lengths of minor nd mjor es re nd respectivel. Further, z z e () e B () Foci re (0, ±e) L S L (c) Directrices: e (d) Vertices re : (0, ±) (e) Length of ltus rectum A L z S B A L z e

4 SOLVED EXAMPLES E.: Find the centre, the eccentricit, the foci, the directrices nd the lengths nd the equtions of the es of the ellipse Sol.: The given eqution cn e written s 5( + ) + 9 ( ) 5( + ) + 9( ) 5 ( ) 9 ( ) 5 Shift the origin to O (, ) X + ; Y X Y...(i) 9 5 This is in stndrd form., 5 e e 9 9 Also e. nd e Now for n ellipse in the stndrd form we hve Centre (0, 0) ; foci (± e, 0) ; directrices ± e ; es 0, 0, lengths of mjor is, length of minor is Now for (i) centre is given X 0, Y 0 + 0, 0 i.e. Centre (, ) Foci re given X ± e, Y 0 i.e. + ± ; 0 i.e., nd, Foci (, ); (, ) The eqution of directrices re given X ± e i.e. + ± 9 7 i.e. ; The eqution of es re given X 0, Y 0 i.e. + 0, 0 i.e., Length of the es eing, i.e. 6, 5.

5 E.: Sol.: If the chord through point nd on n ellipse (d, 0) prove tht tn θ θ tn d d. Eqution of the chord joining points nd is intersects the mjor is t θ θ θ θ cos sin Since (d, 0) lies on it d θ cos θ cos θ θ θ cos θ P( ) O R( d, 0) Q( ) θ θ cos θ θ cos d Appling componendo nd dividendo, we get d d θ cos θ cos θ θ θ cos θ θ θ cos sin cos θ θ θ sin θ cos θ tn θ tn E.: A tngent to the ellipse touches it t the point P in the first qudrnt nd meets the nd es in A nd B respectivel. If P divides AB in the rtio :, find the eqution of the tngent t P. Sol.: Let P ( cos, sin ) : 0 < <...(i) Eqution of tngent t P() is A cos, 0 cos nd B 0, sin sin Now P divides segment AB in the rtio : P, cos sin...(ii) 0, B sin P cos A, 0 B (i) & (ii), we hve: cos ; sin Eqution of tngent t P is +.

6 E.: Sol.: If the tngent drwn t point (t, t): t 0 on the prol is the sme s the norml drwn t point ( 5 cos, sin ) on the ellipse + 5 0, find the vlues of t nd. Eqution of tngent t P(t, t) to is t + t...(i) Eqution of norml t Q is cos 5 sin sin cos...(ii) Equtions (i) & (ii) represent the sme line. Compring the coefficients in equtions (i) & (ii), t cos t 5 sin sin. cos t cot 5 cot 5, t cos 5 cos 5 cos or cos 5 0 sin cos + 5 sin 0 ( cos 0) 5 ( cos ) + cos 0 cos 5 n ± cos 5 where 0 < Corresponding vlues of t re given t cos 5 5 E.5: Show tht the sum of squres of perpendiculrs on n tngent to from points on the minor is, ech of which is t distnce from the centre, is. Sol.: The generl eqution of tngent to the ellipse is m ± m...(i) Let points on minor is e P(0, e) nd Q(0, e) s ( e ) Length of perpendiculr from P on (i) is e m p m e m Similrl p m

7 Hence p + p m m { e + ( m + )} {( ) + m + } E.6: Sol.: Show tht the ngle etween the tngents to the ellipse circle + t their points of intersection in first qudrnt is At the points of intersection of ellipse nd circle, or (where > ), nd the ( ) tn. or nd. P, lies in first qudrnt Eqution of tngent t P to the circle is Its slope is m Eqution of the tngent t P to the ellipse is Its slope is m If is the ngle etween these tngents, then tn m m m m

8 ( ) E.7: An tngent to n ellipse is cut the tngents t the etremities of the mjor is t T nd T. Prove tht the circle on TT s dimeter psses through the foci. Sol.: Let the eqution of the ellipse e The etremities A nd A of the mjor is re A(, 0), A(, 0). Equtions of tngents t A nd A re nd An tngent to the ellipse is cos sin The points of intersection re cos ), ( cos ) T, T, ( sin sin The eqution of circle on TT s dimeter is ( cos ) ( cos ) ( )( + ) + 0 sin sin or + sin ( cos sin ) 0 or + + sin 0 or + sin e 0 Foci S(e, 0) nd S ( e, 0) lie on this circle. E.8: Let d e the perpendiculr distnce from the centre of the ellipse to the tngent t point P on the ellipse. If F nd F re the two foci of the ellipse, show tht Sol.: ( PF PF ). d Eqution of the tngent t the point P(cos, sin ) on the given ellipse is cos θ sin θ d cos θ sin θ

9 or d cos θ sin θ Let M nd M e foot of perpendiculrs from P on lines nd e B definition of ellipse, PF e(pm ) nd PF e(pm )...(i) respectivel. e PF e cos ; PF e e cos e PF.PF e cos + (PF PF ) (PF + PF ) PF.PF () ( e cos + ) From eqution (i), e cos...(ii) cos + ( cos ) d or ( )cos d...(iii) d e cos or d e cos E.9: (PF PF ) d Let ABC e n equilterl tringle inscried in the circle +. Suppose perpendiculrs from A, B, C to the mjor is of the ellipse, ( ), meets the ellipse t P, Q, R respectivel so tht P, Q, R lie on the sme side of mjor is s re the corresponding points A, B, C. Prove tht the normls to the ellipse drwn t the points P, Q, R re concurrent. π π Sol.: Let A, B, C hve coordintes (cos, sin), cos, sin, π π cos, sin respectivel; then, P, Q nd R hve coordintes given : P(cos, sin) A π π Q cos, sin nd π π R cos, sin respectivel. Equtions of normls t P, Q, R to ellipse re cos sin...(i) Q B O P R C

10 cos π π cos sin π sin π...(ii)...(iii) Normls (i), (ii) nd (iii) re concurrent, if determinnts of coefficients is zero. i.e., if sec sec π sec π cosec π cosec π cosec 0 i.e., if sin sin π sin π cos cos π cos π sin θ sin π sin π 0 But sin sin π sin sin + sin cos 0 for ll vlues of Hence ppling R R + R + R in the determinnt on L.H.S, we get ll elements of first row s zeros. The normls re concurrent.

11 OBJECTIVE ASSIGNMENT Choose the correct option in the following :. The vlue of eccentricit of the ellipse is equl to () () / (c). The length of ltus rectum of the ellipse + is (d) none of these () () 6 (c) (d) none of these. The eqution represents n ellipse if 0 () > () < < (c) < () none of these. Eqution of ellipse referred to its es s es of coordintes, which psses through the point (, ) nd (, ) is () () (c) (d) none of these If A nd B re two fied points nd P is vrile point such tht PA + PB, where AB, then the locus of P is () A hperol () A prol (c) An ellipse (d) none of these 6. The point where the line l + m + n 0 m touch the ellipse is l m l m () (l, m) (), (c) n n, (d) none of these n n 7. Chords of n ellipse re drwn through the positive end of the minor is. Then there mid points lie on () circle () prol (c) n ellipse (d) hperol is common tngent to nd. Then the vlue of nd the other common tngent re given (), (), (c), + 0 (d) none of these

12 9. A tngent to the ellipse cuts the coordinte es in A nd B. Then eqution of the locus of the mid point of AB is () () + (c) (d) none of these 0. The prmetric representtion of point on the ellipse whose foci re (, 0) nd (7, 0) nd eccentricit is () ( 8 cos θ, sin θ) () ( cos θ, 8 sin θ) (c) ( 8 cos θ, sin θ) (d) none of these. The curve represented (cos t + sin t), (cos t sin t) is () Prol () Ellipse (c) Hperol (d) Circle. Let E e the ellipse 9 nd C e the circle + 9. Let P nd Q e the points (, ) nd (, ) respectivel. Then, () Q lies inside C ut outside E () Q lies outside oth C nd E (c) P lies inside oth C nd E (d) P lies inside C ut outside E. Let P e point on with foci S nd S. Then the mimum re of tringle PS S is () () e (c) (d) none of these. The point of intersection of tngents drwn t the end points of the ltus rectum of for > 0, lies on () e () e (c) e, (d) none of these 5. If norml t point P(with eccentric ngle ) on intersects it gin t point Q (), then cos () () (c) (d) none of these 6. An ellipse hving foci t (, ) nd (, ) nd pssing through the origin hs eccentricit equl to () 7 () 7 (c) 5 7 (d) 5

13 7. If the eccentricit of ellipse is, then ltus rectum of ellipse is 6 () 5 6 () 0 6 (c) 8 6 (d) none of these 8. Length of the mjor is of ellipse (5 0) + (5 + 5) ( 7) is () 0 () 0 (c) 0 7 (d) 9. The line t meets the ellipse in the rel nd distinct points if nd onl if 9 () t < () t < (c) t > (d) none of these 0. The eccentric ngle of point on the ellipse whose distnce from the focus is 5/ unit is () cos () cos (c) cos (d) none of these. The eccentricit of locus of point (h, k) where (h, k) lies on circle + is () () (c) (d). The slopes of the common tngent of ellipse nd circle + re () ± () ± (c) ± (d) none of these. If tngent of slope of ellipse vlue of is is norml to circle then mimum () () (c) (d) none of these. If touches the ellipse, then eccentric ngle of point of contct is () 6 () (c) (d) 5. Locos of point of intersection of the tngent t the end point of the focl chord of ellipse, ( ) is /n () circle () ellipse (c) hperol (d) pir of stright lines

14 6. An ordinte MP of the ellipse meets the uilir circle t, then locus of point of 5 9 intersection of normls t P nd Q to the respective, curves is () + 8 () + (c) + 6 (d) If the tngent t ponit ( cos, sin ) on the ellipse meets the uillr circle in two points, the chord joining them sutends right ngle t the centre, then eccentricit of ellipse is / () ( cos ) () / sin (c) ( sin ) (d) cos 8. The line m ( ) m m is norml to the ellipse for ll vlues of m elongs to () (0, ) () (0, ) (c) R (d) none of these 9. Numer of points on the ellipse to the ellipse is 6 9 from which pir of perpendiculr tngents re drwn 50 0 () 0 () (c) (d) 0. If mimum distnce of n point on the ellipse + + from its centre e r, then r () + () + (c) 5 (d) MORE THAN ONE CORRECT ANSWERS. In the ellipse () foci re t (, ), (, 9) () e /5 (c) centre is (5, ) (d) mjor is is 6.. The points, where the normls to the ellipse + 7 e prllel to the line is/re () (5, ) () (, 5) (c) (, ) (d) ( 5, ). The product of eccentricities of two conics is unit, one of them cn e /n () prol () ellipse (c) hperol (d) circle 6. If the tngent t the point cos, sin to the ellipse is lso tngent to the circle then equls () () (c) (d) 5

15 5. The eccentric ngle of point on the ellipse 6 t point on the ellipse 6 t distnce units from origin is () () (c) 5 (d) 7 6. P(, ) nd Q(, ), < 0, < 0 e the end points of the ltus rectum of ellipse +. The eqution of the prols with ltus rectum PQ re () () (c) (d) 7. An ellipse intersects the hperol orthogonll. The eccentricities of ellipse is reciprocl of tht of hperol. If the es of ellipse re long the co-rodinte es, then () eq. of ellipse is + () The foci of ellipse re (±, 0) (c) eqution of ellipse is + (d) the foci of ellipse re (±, 0) 8. The locus of the imge of the focus of ellipse ellipse is with respect to n of tngents to the 5 9 () ( ) 00 () ( ) 50 (c) ( ) 00 (d) ( ) If the tngent drwn t point (t, t) on the prol is sme s the norml drwn t point (5 cos, sin ) on the ellipse Then () cos 5 () cos 5 (c) t 5 (d) t 5 0. r r 6 r 6r 5 will represents the ellipse, if r lies in the intervl () (, ) () (, ) (c) (5, ) (d) (, )

16 MISCELLANEOUS ASSIGNMENT Comprehension- AB nd CD re mjor nd minor es of n ellipse respectivel which inscries rectngle PQ RS hving side lengths s 8 nd 6 units. The centre of ellipse eing origin. Answer the following questions:. Which of the following cn not e the verte of given ellipse () (6, 0) () (8, 0) (c) (0, 0) (d) (5, 0). The numer of set of four co-norml points out of A, B, C, D, nd P, Q, R, S is () () (c) 5 (d) 6. If eccentricit of ellipse is / then re of the ellipse is () () 7 (c) (d) none of these Comprehension- If is n ellipse ( > ) we drwn perpendiculrs from oth the foci to n tngent of ellipse.. Then the product of length of drwn from oth the foci to n tngent is equl to () () (c) (d) none of these 5. Eqution of ellipse hving foci (± 5, 0) nd tngent of ellipse is () 9 () 9 (c) 6 9 (d) If S nd S re foci nd S nd S re mirror imge of S nd S. w.r.t. n tngent t P then () S, P & S re colliner () S, P & S re colliner (c) tngent t P isected S PS (d) ll re correct 7. Mtch the eccentricit in column B with the ellipse in column A. A. The ellipse meets the stright line 7 (p) the stright line on the is of nd whose es lies 5 long the es of co-ordintes B. The ellipse in which the ngle etween the stright lines joining the (q) foci to n etremit of minor is is /

17 C. The ellipse whose pir of conjugte dimeter re nd (r) 5 D. The ellipse, the distnce etween its foci is 6 nd minor is is 8 (s) none of these 8. A. If the tngent to the ellipse + 6 t the point P() is norml (p) 0 to the circle then / m e B. The eccentric ngle(s) of point on the ellipse + 6 t (q) cos distnce units from the centric of ellipse is/re C. The ngle of intersection of the ellipse + nd (r) prol + is D. If the norml t point P() to the ellipse 5 intersects it gin (s) 5 t the point Q(), then is INTEGER TYPE QUESTIONS 9. Let e e the eccentricit of the ellipse, if its ltus rectum is hlf of the minor is then e is. 0. If f () + + c + d (,, c, d re rtionl numers) nd roots of f () 0 re eccentricities of prol nd rectngulr Hperol then, + + c + d equl to.. The line + 5 k touches the ellipse , then k 5 is.. If the cute ngle etween pir of tngents drwn to the ellipse from the points (, ) is then the vlue of is. Let e e the eccentricit of the ellipse ( ) ( ), then the vlue of 9e is The numer of integrl vlues of for which three distinct chords drwn from (, 0) to the ellipse + re isected the prol is

18 5. If PSQ is focl chord of ellipse such tht SP 8, then length of SQ is 6. If the eqution (5 ) + (5 ) ( + ) ( + ) represents n ellipse then the numer of integrl vlues of is 7. If the locus of middle point of portion of tngent on the ellipse is k, then vlue of k is included etween the es 8. A r emnting from the point (, 0) is incident on the ellipse t the point P with ordinte. If the eqution of the reflected r fter first reflection of +, then is.

19 PREVIOUS YEAR QUESTIONS. Tngent is drwn to ellipse t ( 7 IIT-JEE/JEE-ADVANCE QUESTIONS such tht sum of intercepts on es mde this tngent is minimum, is () () 6 cos, sin (where (0, )). Then the vlue of. The numer of vlues of c such tht the stright line + c touches the curve is () 0 () (c) (d) infinite.. Let nd e non-zero rel numers. Then, the eqution ( + + c)( ) 0 represents (c) 8 (d) () () (c) (d) four stright lines, when c 0 nd, re of the sme sign two stright lines nd circle, when, nd c is of sign opposite to tht of two stright lines nd hperol, when nd re of the sme sign nd c is of sign opposite to tht of circle nd n ellipse, when nd re of the sme sign nd c is the sign opposite to tht of. Let P(, ) nd Q(, ), < 0, < 0, e the end points of the ltus rectum of the ellipse +. The equtions of prols with ltus rectum PQ re () + + () + (c) + (d) 5. The line pssing through the etremit A of the mjor is nd etremit B of the minor is of the ellipse meets its uilir circle t the point M. Then the re of the tringle with vertices t A, M nd the origin O is () 0 () The norml t point P on the ellipse + 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 (c) 0 (d) 7 0 () 5, 7 () 5 9, (c), 7 (d), 7

20 7. The locus of the orthocentre of the tringle formed the lines ( + p) p + p( + p) 0, ( + q) q + q( + q) 0, nd 0, where p q, is () hperol () prol (c) n ellipse (d) stright line Prgrph Tngents re drwn from the point P(, ) to the ellipse 8. The coordintes of A nd B re touching the ellipse t points A nd B. 9 () (, 0) nd (0, ) () 8 6, 5 5 nd 9 8, 5 5 (c) 8 6, 5 5 nd (0, ) (d) (, 0) nd 9 8, The orthocenter of the tringle PAB is () 8 5, 7 () 8 5, 7 8 (c) 8, 5 5 (d) 8 7, The ellipse : E is inscried in rectngle R whose sides re prllel to the coordinte 9 es. Another ellipse E pssing through the point (0, ) circumscries the rectngle R. The ecentricit of the ellipse E is () () (c) (d). A verticl line pssing through the point (h, 0) intersects the ellipse t the points P nd Q. Let the tngents to the ellipse t P nd Q meet t the point R. If (h) re of the tringle PQR, m ( h) nd / h m ( h), then / h DCE QUESTIONS. If the norml t n point P on the ellipse, meets the es in G nd g respectivel, then PG : Pg () : () : (c) : (d) :

21 . The line l + m + n 0 is norml to the ellipse if ( ) ( ) () () m l n l m n ( ) (c) (d) none of these l m n. If the stright line + c is tngent to the ellipse, c will e equl to 8 () ± () ±6 (c) ± (d) ±8. The curve represented (cos t + sin t), (cos t sin t) is () ellipse () prol (c) hperol (d) circle 5. P is n point on the ellipse whose foci re S nd S. Then SP + SP equls () () 6 (c) 6 (d) 6. Wht is the eqution of the ellipse with focii (±, 0) nd eccentricit? () + 8 () + 8 (c) + 0 (d) The eccentricit of the ellipse is () () (c) (d) none of these 8. The foci of the ellipse 5( + ) + 9( + ) 5, re t () (, ) nd (, 6) () (, ) nd (, 6) (c) (, ) nd (, 6) (d) (, ) nd (, ) 9. Let E e the ellipse 9 nd C e the circle +. Let P nd Q e the points (, ) nd (, ) respectivel. Then () Q lies inside C ut outside E () Q lies outside oth C nd E (c) P lies outside oth C nd E (d) P lies inside C ut outside E 0. Let P e vrile point on the ellipse tringle PF F, then the mimum vlue of A is with foci F nd F. If A is the re of the () e () e (c) e (d) e

22 . In n ellipse the ngle etween the lines joining the foci with the positive end of minor is is right ngle, the eccentricit of the ellipse is () () (c) (d) AIEEE/JEE-MAINS QUESTIONS. In n ellipse, the distnce etween its foci is 6 nd minor is is 8. Then its eccentricit is () 5 () 5. A focus of n ellipse is t the origin. The directri is the line nd the eccentricit is /. Then the length of the semi-mjor is is () / () / (c) 5/ (d) 8/. The ellipse + is inscried in rectngle ligned with the coordinte es, which in turn is inscried in nother ellipse tht psses through the point (, 0). Then the eqution of the ellipse is : (c) (d) 5 () + 6 () (c) (d) Eqution of the ellipse whose es re the es of coordintes nd which psses through the point (, ) nd hs eccentricit /5 is : () () (c) (d) An ellipse is drwn tking dimeter of the circle ( ) + s its semi-minor is nd dimeter of the circle + ( ) 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 () + () + 8 (c) + 8 (d) The locus of the foot of perpendiculr drwn from the centre of the ellipse + 6 on n tngent to it is () ( ) 6 + () ( ) 6 (c) ( + ) 6 + (d) ( + ) 6

23 BASIC LEVEL ASSIGNMENT. Find the eqution of the ellipse whose vertices re (5, 0) nd ( 5, 0) nd foci re (, 0) nd (, 0).. Find the eccentricit of the ellipse Find the equtions of the tngents drwn from the point (, ) to the ellipse Find the eccentric ngle of point on the ellipse t distnce from the centre Otin eqution of chord of the ellipse + 6 which hs (0, 0) s its mid-point. 6. Find the foci of the ellipse 5( + ) + 9( + ) Find the eccentricit of the ellipse if () Length of ltus rectum hlf of mjor is () Length of ltus rectum hlf of minor is 8. Find the condition so tht the line l + m + n 0 m e norml to the ellipse. 9. Find the points where the line + cuts, the ellipse + 5. Otin the eqution of the norml t these points nd show tht these norml include n ngle tn (/5). 0. If the norml t the point P() to the ellipse intersects it gin t the point Q(), show tht cos /.. The common tngent of nd lies in I st qudrnt. Find the slope of the common tngent nd length of the tngent intercepted etween the is.. Find point on the curve + 6 whose distnce from the line + 7, is minimum.. Find the equtions to the normls t the ends of the lter rect of ellipse ech psses through n end of the minor is if e + e. nd prove tht. Find the co-ordintes of those points on the ellipse, tngent t which mke equl ngles with the es. Also prove tht the length of the perpendiculr from the centre on either of these is ( ).

24 5. 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. 6. If F (, 0), F (, 0) nd P is n ponit on the curve then PF + PF 7. If n ellipse, if the line joining focus to the etremities of the minor is mde n equilterl tringle with the minor is, then eccentricit of ellipse is 8. If the chords of contct of tngents from two points (, ) nd (, ) to the ellipse t right ngles, then is equl to re 9. The centre of ellipse ( ) ( ) The locus of mid points of focl chord of ellipse is

25 ADVANCED LEVEL ASSIGNMENT. The tngent nd norml t n point A of n ellipse cut its mjor is in points P nd Q respectivel. If PQ, prove tht the eccentric ngle of the point P is given e cos + cos 0. A circle of rdius r is concentric with the ellipse. Prove tht the common tngent is inclined to the mjor is t n ngle tn r r.. Show tht the locus of the middle points of those chords of the ellipse which re drwn through the positive end of the minor is is. Tngents re drwn from point P to the circle + r so tht the chords of contct re tngent to the ellipse + r. Find the locus of P.. 5. Show tht the tngents t the etremities of ll chords of the ellipse which sutend right ngle t the centre intersect on the ellipse. 6. Find the length of the chord of the ellipse whose middle point is 5 6, The locus of the points of intersection of the tngents t the etremities of the chords of ellipse + 6 which touch the ellipse + is 8. Let P e point on the ellipse, 0 < <. Let the line prllel to -is pssing through P meet the circle + t the point Q such tht P nd Q re on the sme side of the -is. For two positive rel numers r nd s, find the locus of the point R on PQ such tht PR : RQ r : s s P vries over the ellipse. 9. Consider the fmil of circles + r, < r < 5. In the first qudrnt, the common tngent to circle of this fmil nd the ellipse meets the co-ordinte es t A nd B, then find the eqution of the locus of the mid-point of AB.

26 , meets the ellipse 0. A tngent to the ellipse Prove tht the tngents t P nd Q re t right ngles. + in the points P nd Q.. If n ellipse slides etween two perpendiculr stright lines, then locus of its centre is. The line pssing through the etremit A of mjor is nd etremit B of minor is of the ellipse + 9 9, meets the uillr circle t the point M. then re of tringle with vertices t A, M nd the origin O is. The norml t point P on the ellipse, + 6 meets the -is t Q. If M is the mid point of the line segment PQ, then locus of M intersects the ltus rectums of given ellipse t the points.. Tngents drwn from point on the circle + to the ellipse t ngle, then tngents re If P() nd Q re two points on the ellipse, then locus of mid points of PQ is

27 ANSWERS Ojective Assignment. (). (). (c). () 5. (c) 6. (c) 7. (c) 8. (d) 9. () 0. (). (). (d). (). () 5. () 6. (c) 7. () 8. () 9. () 0. (). (c). (). (). () 5. (d) 6. (c) 7. (c) 8. (c) 9. (d) 0. (c). (, ). (,d). (,c). (,d) 5. (,,c,d) 6. (,c) 7. (,) 8. (,c) 9. (,d) 0. (,c) Miscellneous Assignment. (d). (c). (). (c) 5. () 6. (d) 7. A-(s);B-(q);C-(p);D-(r) 8. A-(p), (r); B-(r),(s);C-(p); D-(q) 9. () 0. (0). (5). (). () Previous Yer Questions IIT-JEE/JEE-ADVANCE QUESTIONS. (). (c). (). (,c) 5. (d) 6. (c) 7. (d) 8. (d) 9. (c) 0. (c). 9 DCE QUESTIONS. (c). (). (). () 5. () 6. () 7. () 8. (c) 9. (c) 0. (). () MAINS QUESTIONS. (). (d). (). () 5. (d) 6. (c)

28 Bsic Level Assignment /. 0, + 5. /, /, 5, 7 5. All lines pssing through origin. 6. (, ) & (, 6) 7. () e () ( ) 8.. l m n 9. (, ) &, eq. of normls re + 0 nd (, )., / (,) 0. Advnced Level Assignment e. r ( r s) ( r s) circle (±, ±/7). / 5.