JEE 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 stright line. If two different tngents of = re the normls to =, then (). Minimum distnce etween the curves nd is equl to () 7. Minimum distnce etween the curves nd () 6 5 0 is equl to 5 8 5. Sides of equilterl ABC touch the prol, then point A, B nd C lie on () 6. Length of the focl chord of the ellipse the is, is equl to () sin cos sin cos 7. Eccentricit of the ellipse () 5 6 5 8, tht is inclined t n ngle with cos sin cos sin 8. PQ is chord of the ellipse. If O is the centre of the ellipse nd eccentric ngle of the points P nd Q differ, then re of tringle OPQ is () / /

9. An ellipse with mjor nd minor es of length 0 nd 0 respectivel, slides long the co-ordinte es nd lws remins confined in the first qudrnt. The locus of the centre of the ellipse will e the rc of circle. The length of this rc will e equl to () 0 units 5 units 5 units 5 units 0. Consider circle d nd n ellipse, d > mi. {,} from vrile point P on the circle, tngents PA nd PB re drwn to the ellipse. Locus of the mid point of chord AB is () d d. PQ is vrile chord of the ellipse of the ellipse then, (O eing the origin) is equl to OP OQ d d. if PQ sutend right ngle t the centre (). Consider n chord of the hperol =c tht is prllel to the line =. Circles re drwn hving this chord s dimeter. All these circles will pss through two fied points whose co ordintes re () c,c, c, c c,c, c, c c,c, c, c c,c, c,c. The tngent t point P on the hperol, meets one of its directi t the point Q. If the line segment PQ sutends n ngle t the corresponding focus, thn is lws equl to () 6. If tngent nd norml to the hperol = c, t n point P cuts off intercept nd on the is respectivel nd nd on the is, then is lws equl to () 0 none of these 5. Locus of the mid point of the chord of the hperol, tht touch the prol is ()

6. Locus of the point of intersection of tngent drwn to the hperol etremities of n norml chord is () c 0 c 0 c 0 c 0 c t the 7. A wter jet from fountin reches its mimum height of m t distnce 0.5 m from the verticl pssing through the point O of wter outlet. The height of the jet ove the horizontl OX t distnce of 0.75m from the point O is () 5m 6m m 7m 8. Eqution of norml to the curve = 6 + 6 which is perpendiculr o the stright line joining the origin to the verte of the prol is () = 0 + = 0 = 0 + = 0 9. A circle drwn on n focl chord of the prol = s dimeter cuts prol t two points t nd t (other thn the etrimit of focl chord) the () tt = tt = tt = none of these 0. Two prols with the sme is, focus of ech eing eterior to the other nd the ltus rectum eing nd. The locus of the middle points of the intercepts etween the prols mde on the lines prllel to the common is is () stright line if = prllel line if prol for ll, ellipse if >. If three distinct norml cn e drwn to the prol = 9 from the point (, 0) then rnge of vlues of is () No rel vlues possile (, ) (, ) none of these. If the curves nd for suitle vlue of cut on four conclic points, the eqution () none of these. Angle sutended common tngents of two ellipse t origin is () none of these +5 = 00 nd. If PQR e n equilterl tringle inscried in the uillr circle of the ellipse (>) nd PQR e corresponding tringle inscried within the ellipse then centriod of the tringle PQR lies t () centre of ellipse focus of ellipse etween focus nd centre of mjor is none of these

5. The norml t vrile point P on the ellipse, > of eccentricit e meets the es of the ellipse Q nd R then the locus of the mid point QR is coinc with eccentricit e such tht () e is independent of e e = e = e e = e 6. If vrile line cos sin P, which is chord of the hperol, sutends right ngle t the centre of the hperol then it lws touches fied circle whose rdius is () 7. Let n doule ordinte PNP of the hperol e produced oth sides to 5 6 meet the smptotes in Q nd Q, then PQ. PQ is equl to () 5 6 none of these 8. The eqution of the line of ltus rectum of the rectngulr hperol = c is () c c c 0 9. The line prllel to the norml to the curve = is/re () 5 0 5 0 5 0 5 0 0. The line p p p for different vlues of p touches () An ellipse of eccentricit Hperol of eccentricit An ellipse of eccentricit none of these. If, re the eccentric ngels of the ends of focl chord of the ellipse then the eccentricit of the ellipse is sin sin () sin cos cos sin sin sin cos cos. If chords of contct of tngents from two points, nd () re t right ngle, then cos cos,, to the ellipse

. PM nd PN re the perpendiculrs from n point P on the rectngulr hperol = c to the smptotes. The locus of the mid point of MN is hperol with eccentricit (). An ellipse hs eccentricit nd one focus t s,. Its one directi is the common tngent, (nerer to S) to the circle nd. The eqution of the ellipse in stndrd form is () 9 9 9 5. If p nd p re the perpendiculrs from the origin on the stright lines sec cosec nd cos sin cos, then () p p p p p p p p 6. If c is the centre nd A, B re two points on the conic 9 8 6 0 such tht ACB, then CA CB is equl to 6 () 6 6 6 7. A point moves such tht the sum of the squres of its distnces from the two sides of length of rectngle is twice the sum of the squres of its distnces from the other two sides of length. the locus of the point cn e () circle n ellipse hperol 8. If the tngent t the point pir of lines P, to the prol meets the prol t Q nd R, then the mid point of QR is (),,,, 9. If, re the eccentric ngels of the etremities of focl chord of n ellipse, then the eccentricit of the ellipse is cos cos sin sin () cos sin

cos cos cos sin sin sin 0. PQ nd RS re two perpendiculr chords of t rectngulr hperol = c. If O is the centre of the hperol, then the product of the slopes of OP, OQ, OR nd OS is equl to (). Let f e the focus of the prol. From the end point (P) of focl chord PF perpendiculr PM is drwn to directi. From P line is drwn through the mid point (R) of FM, then the ngle etween PR nd FM is () 5 60 90 none of these. A norml drwn to prol meet the curve gin t Q such tht ngle sutend PQ t verte is 90 then coordinte of P cn e () 8, 8,, none of these. If the focus of prol ( k) =( h) lws lies etween the line + = nd + = then () 0 h k 0 h k h k h k. If = m e rectngulr hperol whose rnches lies onl in the nd nd th qudrnt then () m m m not possile 5. A tngent to the ellipse t n point P meet the line = 0 t point Q. Let 5 6 R the imge of Q in the line =, then circle whose etremities of dimeter re Q nd R psses through fied point. The fied point is (,0) (B) (5,0) (0,0) (D) (,0) 6. Numer of points on the ellipse from which pir of perpendiculr tngents 50 0 re drwn to the ellipse is 6 9 0 (B) (D) 7. An ellipse is drwn with mjor nd minor es of lengths 0 nd 8 respectivel. Using one focus nd centre, circle is drwn tht is tngent to the ellipse, with no prt of the circle eing outside the ellipse. The rdius of the circle is : (B) (D) 8. A circle hs the sme centre s n ellipse nd psses through the foci F nd F of the ellipse, such tht the two curves intersect in points. Let P e n one of their point of intersection. If the mjor is of the ellipse is 7 nd the re of the tringle PF F is 0, then the distnce etween the foci is (B) (D) 5

9. An ordinte MP of n ellipse meets the uilir circle in Q, then locus 5 9 of point of intersection of normls t P nd Q to the respective curves, is (B) (D) 5 50. Numer of distinct norml lines tht cn e drwn to ellipse from the 69 5 point P(0,6) is one (B) two three (D) four 5. If PQ is focl chord of ellipse which psses through S (,0) nd PS = 5 6 then length of chord PQ is 8 (B) 6 0 (D) 5. If P is moving point in the -plne is such w tht perimeter of tringle PQR is 6 {where Q (, 5), R (7, 5 )} then mimum re of tringle PQR is 6 sq. unit (B) sq. unit 8 sq. unit (D) 9 sq. unit 5. If f() is decresing function then the set of vlues of k, for which the mjor is of the ellipse f k k 5 f k is the -is, is k, (B) k, k,, (D) k,, 5. The eqution to the locus of the middle point of the portion of the tngent to the ellipse included etween the co-ordinte es is the curve 6 9 9 + 6 = (B) 6 + 9 = + = (D) 9 + 6 = 55. From point P(,) pir of tngent s re drwn to hperol H in which one tngent to ech re of hperol. H re 5 0 nd 0 0 then eccentricit of H is (B) (D) 56. If vrile lines hs its intercepts on the coordintes es e, e where e, e re the eccentricities of hperol nd its conjugte hperol, then the line lws touches the circle r, where r = (B) (D) cn not e decided 57. If ngle etween smptote s of hperol is 0 nd product of perpendiculrs drwn from foci upon its n tngent is 9, then locus of point of intersection of perpendiculr tngent of the hperol cn e 6 (B) 9

(D) 8 58. C e curve which is locus of point of intersection of lines = + m nd m= m. A circle s 5 intersects the curve C t four points, P, Q, R nd S. If O is centre of the curve C, then OP + OQ + OS is 50 (B) 00 5 (D) 5/ 59. The comined eqution of the smptotes of the hperol 5 5 0 is 5 5 0 (B) 5 5 0 5 0 (D) none of these 60. If then the chord joining the points nd for the hperol psses through focus (B) centre one of the end points of the trnsverse is (D) one of the end points of the conjugtes is 6. For given non-zero vlue of m ech of the lines m nd m meets the hperol = t point. Sum of the ordintes of these points, is m m (B) m m 0 (D) m 6. The eqution of the trnsverse is of the hperol (-) + (+) = ( + ) is + = 0 (B) + = 9 = (D) + = 0 6. For which of the hperol, we cn hve more thn one pir of perpendiculr tngents? (B) 9 9 (D) 6. From point (,) tngents re drwn to the hperol then point of contct 6 9 lie in I nd II qudrnts (B) I nd IV qudrnts I nd III qudrnts (D) III nd IV qudrnts 65. A circle is descried whose centre is the verte nd whose dimeter is three-qurters of the ltus rectum of the prol. If PQ is the common chord of the circle nd the prol nd L L is the ltus rectum, then the re of the trpezium PL L Q is (B) (D) 66. From the point (5,) three normls re drwn to the prol, then centroid of tringle formed three-co-norml points is

6,0 (B),0 6,0 (D) 6,0 67. Through the verte O of the prol = two chords OP nd OQ re drwn nd the circles on OP nd OQ s dimeter intersect in R. If, & re the ngle mde with the is the tngents t P nd Q on the prol nd OR then cot cot is equl to tn (B) tn 0 (D) cot 68. A r of light trvels long line = nd strikes the surfce of curve then eqution of the line long reflected r trvel is = 0 (B) = + = (D) + = nd M is the foot of perpendiculr drwn from P on the directi of the prol, then length of the ech side of n equilterl tringle SMP, where S is focus of the prol, is (B) 69. If P e point on the prol 6 (D) 8 70. If the locus of middle point of contct of tngent drwn to the prol foot of perpendiculr drwn from its focus to the tngent is conic then length of ltusrecturm of this conic is 9/ (B) 9 8 (D) 9/ 7. Normls t three points P, Q, R t the prol focus, if SP. SQ. SR SA, then is equl to (B) (D) 7. If the chord of contct of tngents from point P to the prol prol, the locus of P is circle (B) prol ellipse (D) hperol 7. Minimum re of circle which touches the prol s 8 nd meet in point A nd S e its nd 7. Let P nd Q e points (,-) nd (9,6) of the prol touches the is 9 9 sq.unit (B) sq.unit 6 9 9 sq.unit (D) sq.unit 8. Let R e point on the rc of the prol etween P nd Q. Then the re of PRQ is lrgest when PRQ 90 (B) then point R is (,) the point R is, (D) none of these ONE OR MORE THAN ONE CORRECT

75. If P is point of the ellipse PSS, then PS PS, if > (B) PS PS, if < e tn tn e, whose foci re S nd S. Let PSS nd (D) tn tn when > 76. If the chord through the points whose eccentric ngles re nd on the ellipse, psses through focus, then the vlue of tn / tn / is : e e (B) e e e e (D) e e 77. The eqution 8 6 c cnnot represent rel pir of stright lines for n vlue of c (B) represents n ellipse, if c > 0 represents empt set, if c < 0 (D) point, if c = 0 78. If foci of concide with the foci of nd eccentricit of the 5 9 hperol is, then 6 (B) there is no director circle to the hperol centre of the director circle is (0,0) (D) length of ltus rectum of the hperol = 79. If (5,) nd (,7) re the foci of conic pssing through the origin then the eccentricit of conic is 86 86 5 80. For the hperol One of the directri is (B) 86 (D) 86 8 9 6 8 5 0 (B) length of ltus rectum = 9 5 Focii re (6,) nd (-,) (D) eccentricit is 5 Sujective Tpe 8. Two prols hve common is nd concvities in opposite directions; if n line prllel to the common is meet the prols in P nd P, prove tht the locus of the middle point of PP is nother prol, provided tht the ltus rect of the given prols re unequl.

8. The norml t n point P meets the is in G nd the tngent t the verte in G, if A e the verte nd the rectngle AGQG e completed, prove tht the eqution to the locus of Q is = + 8. If norml to prol mke n ngle of with the is, show tht it will cut the curve gin t ngle tn tn 8. If PQ e norml chord of the prol nd if S e the focus, prove tht the locus of the centroid of the tringle SPQ is the curve 6 5 8 8 85. If from the verte of prol pir of chord e drwn t right ngles to one nother nd with these chords s djcent sides rectngle e mde, prove tht the locus of the further ngle of the rectngle is the prol =( 8) 86. Prove tht the orthocentres of the tringles formed three tngents nd the corresponding three normls to prol re equidistnt from the is. 87. Circles re drwn through the verte of the prol to cut the prol t the other point of intersection. Prove tht the locus of the centres of the circles is the curve 88. Through the verte A of the prol -= two chords AP nd AQ re drwn, nd the circles on AP nd AQ s dimeters intersect in R. Prove tht, if, nd e the ngles mde with the is the tngents t P nd Q nd AR, then cot cot tn 0 89. If the normls t the three points P,Q nd R meet in point nd if PP, QQ nd RR e chords prllel to QR, RP nd PQ respectivel, prove tht the nromls t P, Q nd R nd lso meet in point 90. The sides of tringle touch prol nd two of its ngulr point lie on nother prol with its is in the sme direction; prove tht the locus of the third ngulr points in nother prol. 9. The tngent t n point P of circle meets the tngent t fied point in T, nd T is joined to B, the other end of the dimeters through A; prove tht the locus of the intersection of AP nd BT is n ellipse whose eccentricit is 9. Find the locus of intersection of the two stright lines t t 0 nd t 0. Prove lso tht the meet t the point whose eccentric ngle is tn t. 9. If the stright line =m+c meet the ellipse, prove tht the eqution to the circle, descried on the line joining the points of intersection s dimeter, is m m c c c m 0. 9. Prove tht the sum of the eccentric ngles of the etremities of chord, which is drwn in given direction, is constnt nd equl to twice the eccentric ngle of the point t which the tngent is prllel to the given direction. 95. The eccentric ngles of two point P nd Q on the ellipse re nd ; prove tht the re of the prllelogrm formed the tngents t the ends of the dimeters. Through P nd q is cosec nd hence tht it is lest when P nd q re t the end of conjugte dimeters.

96. In oth n ellipse nd hperol, prove tht the focl distnce of n point nd the perpendiculr from the centre upon the tngent s it meet on circle whose centre is the focus nd whose rdius is the semi trnsverse is. 97. Given the se of tringle nd the rtio of the tngent of hlf the se ngles, prove tht the verte moves on hperol whose foci re etremities of the se. 98. A stright line is drwn prllel to the conjugte is of hperol to meet it nd the conjugte hperol in the point s P nd q; show tht the tngents t P nd Q meet on the curve 99. Find the eqution to the hperol, whose smptotes re stright lines ++=0 nd ++5=0, nd whish psses through the point (, ). Write down lso the eqution to the conjugte hperol. 00. C is the centre of the hperol nd the tngent t n point P meets the smptote in the point q nd r. Prove tht the eqution to the locus of the centre of the circle circumscriing the tringle CQR is 0. Let V e the verte nd L e the ltusrectum of the prol. then the eqution of the prol whose verte is t V, ltusrectum is L/ nd is is perpendiculr to the is of the given prol. (B) (D) 0. If eqution of tngent t P, Q nd verte A of prol re 7 0, + -0=0 nd = 0 respectivel, then focus is (,5) (B) length of ltus rectum is is is 9 0 9 9 (D) verte is, 0. If A nd B re points on the prol with verte O such tht OA perpendiculr to OB nd hving length, r nd r respectivel, then the vlue of / / r r / / r r is 6 (B) (D) none of these 0. Let P, Q nd R re three co-norml points on the prol. then the correct sttement(s) is / re lgeric sum of the slopes of the normls t P, Q nd R vnishes (B) lgeric sum of the ordintes of the points P, Q nd R vnishes centroid of the tringle PQR lies on the is of the prol (D) circle circumscriing the tringle PQR psses through the verte of the prol. 05. The locus of the mid point of the focl rdii of vrile point moving on the prol, is prol whose ltus rectum is hlf the ltus rectum of the originl prol (B) verte is (/.0) directi is -is (D) Focus hs the co-ordintes (,0)