/ 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


 Ashlie Nicholson
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1 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 ellipse x re equl nd equl to whose distnce from the center of the ellipse. Eccentricit of this ellipse is equl to. If the line = mx + c, interests the ellipse x, t points whose eccentric ngles differ /, then ( m + ) = 4c ( + m ) = 4c m + = 4c + m = 4c 4. Consider n ellipse with mjor nd minor xes of length 0 nd 8 units respectivel. The rdius of lrgest circle tht cn e inscried in this ellipse, it is given tht centre of this circle is one focus of the ellipse, is equl to 4 units units 6 units none of these 5. Eccentricit of the ellipse 5x + 6x + 5 = 8 is 6. The tngent t the point '' on the ellipse x + = meets the uxilir circle in two points which sutends right ngle t the centre, then the eccentricit 'e' of the ellipse is given the eqution e ( + cos ) = e. (cosec ) = e ( + sin ) = e ( + tn ) = 7. S nd T re the foci of n ellipse nd B is n end of the minor xis. If STB is equilterl, then e is
2 /4 / / none of these 8. A ldder units long slides in verticl plne with its ends in contct with verticl wll nd horizontl floor long xxis. The locus of point on the ldder 4 units from its foot hs the eqution : x 4 + = x = 6 64 x = x = 9. Eccentric ngle of point on the ellipse x + = 6 t distnce units from the centre of the ellipse is / / / 4 none of these 0. The point of intersection of the tngents t the point P on the ellipse x corresponding point Q on the uxilir circle meet on the line : x = /e x = 0 = 0 none of these. If nd re eccentric ngles of the ends of focl chord of the ellipse x tn tn e e e e is equl to e e e e = nd its, then. The distnces from the foci of P(, ) on the ellipse x re none of these 5. If tn. tn = x then the chord joining two points & on the ellipse
3 will sutend right ngle t focus end of the mjor xis centre end of the minor xis 4. The equtions of the common tngents to the ellipse, x + 4 = 8 & the prol = 4x re x = ± x 4 = ± x + 4 = ± none of these 5. If O is the centre, OA the semimjor xis nd S the focus of n ellipse, the eccentric ngle of n point P is POS PSA PAS none of these 6. If A nd B re two fixed points nd P is vrile point such tht PA + PB = 4, the locus of P is prol n ellipse hperol none of these F HG 7. If P( ) nd Q I K J re two points one the ellipse x x x 4, locus of midpoint of PQ is x none of these 8. The length of the chord of the ellipse x where midpoint is F HG, I K J none of these 9. The sum of the squre of perpendiculrs on n tngent to the ellipse x from two point on the minor xis, ech t distnce re from the centre, is If ltus rectum of the ellipse x tn sec is / then ( 0 ) is equl to / / 6 / 8 none of these
4 SET II. The length of the mjor xis of the ellipse (5x 0) + (5 + 5) = (x 4 7) 4 4 is. The tngent nd norml to the ellipse x + 4 = 4 t point P() on it meets the mjor xes in Q nd R respectivel. If QR =, then cos is equl to 4 5 none of these. x The ellipse nd the stright line = mx + c intersect in rel points onl if m < c m > c m c c 4. The foci of the ellipse 5 (x + ) + 9 ( + ) = 5 re t (, ) nd (, 6) (, ) nd (, 6) (, ) nd (, ) (, ) nd (, 6). 5. The prmetric representtion of point on the ellipse whose foci re (, 0) nd (7, 0) nd eccentricit / is ( + 8cos, 4 sin ) (8cos, 4 sin ) ( + 4 cos, 8sin ) none of these 6. x The eqution =, will represent n ellipse if 6 (, ) (, 6) (, ) (6, ) (, 6) ~ {4} 7. Tngents re drwn to the ellipse x + = 4 from n ritrr point on the line x + = 4, the corresponding chord of contct will lws pss through fixed point, whose coordintes re,, 8. The line = x touches the ellipse x + 4 =, t,,
5 , (, ) (, ) None of these x 9. The norml drwn to the ellipse t the extremit of the ltus rectum psses through the extremit of the minor xis. Eccentricit of this ellipse is equl to 5 5 x 0. The line 5x = 8 is norml to the ellipse. If e the eccentric ngle of the foot 5 9 of this norml, then is equl to None of these 6 4 x. Tngent drwn to the ellipse t point P meets the coordinte xes t points A nd B respectivel. Locus of midpoint of segment AB is x x 4 4 x x x. Tngents PA nd PB re drwn to the ellipse from the point P(0, 5). Are of tringle 6 9 PAB is equl to sq. units sq. units sq. units sq. units x. Tngents re drwn to the ellipse from n point on the prol = 4x. The 6 9 corresponding chord of contct will touch prol, whose eqution is + 4x = 0 = 4x + 9x = 0 = 9x 4. The norml t vrile point P on n ellipse x = of eccentricit e meets the xes of the ellipse in Q nd R then the locus of the midpoint of QR is conic with n eccentricit e such tht e is independent of e e = e = e e = /e
6 5. An ellipse is such tht the length of the ltus rectum is equl to the sum of the lengths of its semi principl xes. Then Ellipse ulges to circle Ellipse ecomes line segment etween the two foci Ellipse ecomes prol none of these 6. If the line x + 4 = 7 touches the ellipse x + 4 = then, the point of contct is 7, 7, 7, 7 none of these 7. A common tngent to 9x + 6 = 44 ; x + 4 = 0 & x + x + = 0 is = x = 4 x = 4 = 8. If F & F re the feet of the perpendiculrs from the foci S & S of n ellipse x = on the 5 tngent t n point P on the ellipse, then (S F ). (S F ) is equl to The re of the rectngle formed the perpendiculrs from the centre of the stndrd ellipse to the tngent nd norml t its point whose eccentric ngle is /4 is 0. If & re the eccentric ngles of the extremities of focl chord of n stndrd ellipse, then the eccentricit of the ellipse is : cos cos cos( ) sin sin sin( ) cos cos cos( ) sin sin sin( )
7 SET III Multiple choice questions with one or more thn one correct choice. If x = 0 is common tngent to = 4x & x 4 =, then = x = 0 x + 4 = 0 =. If numer of ellipse e descried hving the sme mjor xis ut vrile minor xis then the tngents t the ends of their ltusrectum pss through fixed points. Then the fixed points is/re (0, ) (, ) (0,  ) (0, 0). Eccentric ngle of point on the ellipse x + = 6 t distnce units from the centre of the ellipse is For the ellipse x + 4 6x = 0 centre is (, ) eccentricit is foci re (, ) nd (, ) ll of these re true x 5. If pir of tngents re drwn to the ellipse + = from point P so tht the tngents re 6 9 t right ngles to ech other then the possile coordintes of the point P is/re (, 7) (5, 0) (, 4) ( 5, 5) Red the following pssge nd nswer the questions : x Consider the ellipse ( ) nd circle x + = r. Now n tngent of ellipse will e mx m nd n tngent of circle will e mx r m. 6. The rnge of r for which 4 distinct common tngents re possile [, ] (, ) (, ] [, ) 7. The eqution of common tngent in 4 th qudrnt will e r x r r r r x r r r
8 r x r r r r x r r r 8. Are of qudrilterl formed ll the common tngent will e r ( ) ( r ) (r ) r ( ) ( ) ( r ) 4r ( ) ( r ) (r ) r ( ) ( r ) (r ) Red the pssge given elow nd nswer the questions : Suppose tht n ellipse nd circle re respectivel given the eqution x...(i) nd x + + gx + f + c = 0...(ii) x The eqution (x gx f c) 0...(iii) Represents curve which psses through the common points of the ellipse (i) nd the circle (ii). We cn choose so tht the eqution (iii) represents pir of stright lines. In generl we get three vlue of indicting three pir of stright lines cn e through the points. Also when (iii) x represents pir of stright lines the re prllel to the lines (x ) 0, which represents pir of lines equll inclined to xes (the term contining x is sent). Hence two stright lines through the points of intersection of n ellipse nd n circle mke equl ngles with the xes. Aove description cn e pplied identicll for hperol nd circle. x 9. The rdius of the circle pssing through the points of intersection of ellipse nd x = 0 is x 0. If,,, e eccentric ngles of the four concclic points of the ellipse, then is equl to
9 (n ) (n ) n n x. Let the eccentric ngles of three points P, Q nd R on the ellipse re, nd. A circle through P, Q nd R cuts the ellipse gin t S, then the eccentric ngle of S is. Suppose two lines re drwn through the common points of intersection of hperol x nd circle x + + gx + f + c = 0. If these lines re inclined t ngle nd to x xis then, tn. The numer of pir of stright line formed points of intersection of rectngulr hperol x = nd circle x + 4x 5 = 0 is 0 Red the pssge given elow nd nswer the questions : If PCP e dimeter of the ellipse nd the dimeter DCD e drwn prllel to the tngents t P nd P, then PCP e prllel to the tngents t D nd D. Two such dimeters re known s conjugte dimeters. Condition tht the lines = mx, = x is mm. If = n odd multiple of. mx should lie long conjugte dimeters of the ellipse x 4. Tngents t the extremities of conjugte dimeters of the ellipse intersect on the ellipse
10 x x x = none of these 5. The locus of the intersection of normls t the extremities of conjugte dimeters of the ellipse 6. If x = is ( x + ) = ( ). ( x ) ( x ) = ( ). ( x + ) ( x ) = ( + ). ( x + ) none of these = x = e n ellipse referred to two conjugte dimeters s xes the lines = mx, m x will e conjugte dimeters if mm mm mm none of these 7. CP nd CDR conjugte semidimeters of n ellipse nd the tngent t P meets n other pirs of conjugte dimeters in T nd T, then TP.PT CD TP.PT CD TP.PT CD none of these 8. True or Flse (i) Given the se of tringle nd sum of its sides then the locus of the centre of its in circle is n ellipse. x (ii) If tngent of slope m t point of the ellipse psses through (, 0) nd if e denotes the eccentricit of the ellipse, then m + e =. (iii) (iv) (v) If the ltus rectum of n ellipse is equl to hlf the minor xis, then its eccentricit is equl to 4. D A line of fixed length ( + ) moves so tht its ends re lws on two fixed perpendiculr stright lines. The locus of the point which divided this line into portions of lengths & is n ellipse An ellipse slides etween two perpendiculr stright lines. Then the locus of its centre is n ellipse
11 9. Fill in the lnks : (i) The eccentricit of n ellipse whose ltus rectum equls hlf its mjor xis is. (ii) The equtions of the common tngents to the ellipse, x + 4 = 8 & the prol = 4x re &. (iii) The eqution of the ellipse with its centre t (, ), focus t (6, ) nd pssing through the point (4, 6) is. (iv) P & Q re corresponding points on the ellipse x 6 + =, nd the uxilir circle respectivel. 9 The norml t P to the ellipse meets CQ in R where C is centre of the ellipse. Then l (CR) = (v) The sum of the squres of the reciprocls of two perpendiculr dimeters of the ellipse, 5x + 4 = is equl to. 0. Mtch the column Column I Column II () The length of the semi ltus rectum of n ellipse is one third of its mjor xis, then its eccentricit would e (P) xis () The point from which the tngents to the ellipse 5x + 4 = 0 re perpendiculr, is (Q) (, ) (c) (d) (e) An ellipse hs OB s semi minor xis. F, F re its foci nd the ngle FBF is right ngle. Then the eccentricit of the ellipse is (R), (x ) The centre of the ellipse 6 (x ) + is 9 The eqution of norml t the point (0, ) of the ellipse 9x + 5 = 45 is (S) (T) d i
12 SET I. D. C. A 4. D 5. B 6. C 7. A 8. C 9. C 0. C. B. C. B 4. C 5. D 6. B 7. A 8. D 9. A 0. A SET II. B. B. C 4. A 5. A 6. D 7. A 8. D 9. A 0. C. C. B. C 4. C 5. A 6. D 7. C 8. B 9. A 0. D SET III. AB. AC. AC 4. AB 5. ABCD 6. B 7. C 8. A 9. B 0. C. C. C. C 4. B 5. A 6. B 7. B 8. (i) T (ii) T (iii) F (iv) T (v) F 9. (i) e = (ii) x + 4 = ± (iv) 7 units (v) 9/4 (iii) ( x ) ( ) S, R, ct, dq, ep
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