, MATHS H.O.D.: SUHAG R.KARIYA, BHOPAL, CONIC SECTION PART 8 OF

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1 DOWNLOAD FREE FROM PH.: , Some questions (Assertion Reson tpe) re given elow. Ech question contins Sttement (Assertion) nd Sttement (Reson). Ech question hs 4 choices (A), (B), (C) nd (D) out of which ONLY ONE is correct. So select the correct choice : Choices re : (A) Sttement is True, Sttement is True; Sttement is correct eplntion for Sttement. (B) Sttement is True, Sttement is True; Sttement is NOT correct eplntion for Sttement. (C) Sttement is True, Sttement is Flse. (D) Sttement is Flse, Sttement is True. PARABOLA 86. Sttement- : Slope of tngents drwn from (4, 0) to prol = 9 re, Sttement- : Ever prol is smmetric out its directri. 87. Sttement- : Though (λ, λ + ) there cn t e more thn one norml to the prol = 4, if λ <. Sttement- : The point (λ, λ + ) lies outside the prol for ll λ. 88. Sttement- : If + = k is norml to the prol =, then k is 9. Sttement- : Eqution of norml to the prol = 4 is m + m + m 3 = Sttement- : If, k re the segments of focl chord of the prol = 4, then k is equl to /-. Sttement- : Ltus rectum of the prol = 4 is H.M. etween the segments of n focl chord of the prol 90. Sttement- : Two prols = 4 nd = 4 hve common tngent + + = 0 Sttement- : + + = 0 is common tngent to the prols = 4 nd = 4 nd point of contcts lie on their respective end points of ltus rectum. 9. Sttement- : In prol = 4, the circle drwn tking focl rdii s dimeter touches -is. Sttement- : The portion of the tngent intercepted etween point of contct nd directi sutends 90 ngle t focus. 9. Sttement- : The joining points (8, -8) & (/, ), which re ling on prol = 4, pss through focus of prol. Sttement- : Tngents drwn t (8, -8) & (/, -) on the prol = 4 re perpendiculr. 93. Sttement- : There re no common tngents etween circle = 0 nd prol =. Sttement- : Eqution of tngents to the prol = 4 is = m + /m where m denotes slope of tngent. 94. Sttement- : Three distinct normls of the prol = cn pss through point (h,0) where h > 6. Sttement- : If h > then three distinct nromls cn pss through the point (h, 0) to the prol = Sttement- : The normls t the point (4, 4) nd, 4 of the prol = 4 re perpendiculr. Sttement- : The tngents to the prol t the nd of focl chord re perpendiculr. 96. Sttement- : Through (λ, λ + ) there cnnot e more thn one-norml to the prol = 4 if λ <. Sttement- : The point (λ, λ + ) lines out side the prol for ll λ. 97. Sttement- : Slope of tngents drwn from (4, 0) to prol = 9 re /4, 9/4 Sttement- : Ever prol is smmetric out its is. 98. Sttement- : If prol is defined n eqution of the form = + + c where,, c R nd > 0, then the prol must possess minimum. Sttement- : A function defined n eqution of the form = + + c where,, c R nd 0, m not hve n etremum Sttement- : The point (sin α, cos α) does not lie outside the prol + = 0 when,, α π 6 Sttement- : The point (, ) lies outside the prol = 4 if 4 > Sttement- : The line = + touches the prol = 4( + ). Sttement- : The line = m + c touches = 4( + ) if c = m + /m. 30. Sttement- : If PQ is focl chord of the prol = 3 then minimum length of PQ = 3. Sttement- : Ltus rectum of prol is the shortest focl chord. 30. Sttement- : Through (λ, λ + ), there cn t e more thn one norml to the prol = 4 if λ <. Sttement : The point (λ, λ + ) lies outside the prol for ll λ R ~ {} Sttement : Perpendiculr tngents to prol = 8 meets on + = 0 Sttement : Perpendiculr tngents of prol meets on tngent t the verte. 8 of 9

2 DOWNLOAD FREE FROM PH.: , Let = 4 nd = 4 e two prols Sttement-: The eqution of the common tngent to the prols is + + = 0 Sttement-: Both the prols re reflected to ech other out the line = Let = 4 ( + ) nd = 4 ( + ) re two prols Sttement- : Tngents re drwn from the locus of the point re mutull perpendiculr Stte.-: The locus of the point from which mutull perpendiculr tngents cn e drwn to the given com is + + = 0 ELLIPSE 306. Tngents re drwn from the point (-3, 4) to the curve = 44. STATEMENT -: The tngents re mutull perpendiculr. STATEMENT-: The locus of the points from which mutull perpendiculr tngents cn e drwn to the given curve is + = Sttement : Circle + = 9, nd the circle ( 5) ( 3) + ( ) = 0 touches ech other internll. Sttement : Circle descried on the focl distnce s dimeter of the ellipse = 36 touch the uilir circle + = 9 internll 308. Sttement : If the tngents from the point (λ, 3) to the ellipse = re t right ngles then λ is equl to ±. Sttement : The locus of the point of the intersection of two perpendiculr tngents to the ellipse + =, is + = Sttement : 5 = 0 is the eqution of the tngent to the ellipse = 44. Sttement : The eqution of the tngent to the ellipse + = is of the form = m ± m Sttement : At the most four normls cn e drwn from given point to given ellipse. Sttement : The stndrd eqution + = of n ellipse does not chnge on chnging nd. 3. Sttement : The focl distnce of the point ( 4 3, 5 ) on the ellipse = 600 will e 7 nd 3. Sttement : The rdius of the circle pssing through the foci of the ellipse its centre t (0, 3) is Sttement- : The lest vlue of the length of the tngents to Sttement- : If nd e n two positive numers then + = nd hving = intercepted etween the coordinte es is Sttement- : In n ellipse the sum of the distnces etween foci is lws less thn the sum of focl distnces of n point on it. Sttement- : The eccentricit of n ellipse is less thn. 34. Sttement- : An chord of the conic + + =, through (0, 0) is isected t (0, 0) Sttement- : The centre of conic is point through which ever chord is isected. 35. Sttement- : A tngent of the ellipse + 4 = 4 meets the ellipse + = 6 t P & Q. The ngle etween the tngents t P nd Q of the ellipse + = 6 is π/ Sttement- : If the two tngents from to the ellipse / + / = re t right ngle, then locus of P is the circle + = of 9

3 DOWNLOAD FREE FROM PH.: , Sttement- : The eqution of the tngents drwn t the ends of the mjor is of the ellipse = 0 is = 0, = 7. Sttement- : The eqution of the tngent drwn t the ends of mjor is of the ellipse / + / = lws prllel to -is 37. Sttement- : Tngents drwn from the point (3, 4) on to the ellipse + = will e mutull perpendiculr 6 9 Sttement- : The points (3, 4) lies on the circle + = 5 which is director circle to the ellipse + = Sttement- : For ellipse + =, the product of the perpendiculr drwn from focii on n tngent is Sttement- : For ellipse + =, the foot of the perpendiculrs drwn from foci on n tngent lies on the circle = 5 which is uilir circle of the ellipse. 39. Sttement- : If line + = 3 is tngent to n ellipse with foci (4, 3) & (6, ) t the point (, ), then = 7. Sttement- : Tngent nd norml to the ellipse t n point isects the ngle sutended foci t tht point. 30. Sttement- : Tngents re drwn to the ellipse + = t the points, where it is intersected the line + 3 =. 4 Point of intersection of these tngents is (8, 6). Sttement- : Eqution of chord of contct to the ellipse + = from n eternl point is given + = 0 3. Sttement- : In n ellipse the sum of the distnces etween foci is lws less thn the sum of focl distnces of n point on it. Sttement- : The eccentricit of n ellipse is less thn. 3. Sttement- : The eqution + + λ = 0 cn never represent hperol Sttement- : The generl eqution of second degree represent hperol it h >. 33. Sttement- : The eqution of the director circle to the ellipse = 36 is + = 3. Sttement- : The locus of the point of intersection of perpendiculr tngents to n ellipse is clled the director circle. 34. Sttement- : The eqution of tngent to the ellipse = 36 t the point (3, ) is =. 3 Sttement- : Tngent t (, ) to the ellipse + = is = 35. Sttement- : The mimum re of PS S where S, S re foci of the ellipse + = nd P is n vrile point on it, is e, where e is eccentricit of the ellipse. Sttement- : The coordintes of pre ( sec θ, tn θ). 36. Sttement- : In n ellipse the sum of the distnces etween foci is lws less thn the sum of focl distnce of n point on it. Sttement- : The eccentricit of ellipse is less thn. 37. Let Y = ± 9 3 [3, ) nd Y = ± HYPERBOLA 9 3 e (-, -3] two curves. Sttement : The numer of tngents tht cn e drwn from Y = ± 9 3 is zero 0 5, 3 to the curve 84 of 9

4 DOWNLOAD FREE FROM PH.: , Sttement : The point 5, 3 lies on the curve Y = ± Sttement : If (3, 4) is point of hperol hving focus (3, 0) nd (λ, 0) nd length of the trnsverse is eing unit then λ cn tke the vlue 0 or 3. Sttement : S P SP =, where S nd S re the two focus = length of the trnsverse is nd P e n point on the hperol. 39. Sttement : The eccentricit of the hperol = 0 is 5 4. Sttement : The eccentricit of the hperol = is equl to 330. Let,, α R {0}, where, re constnts nd α is prmeter. Sttement : All the memers of the fmil of hperols smptotes = hve the sme pir of α Sttement : Chnge in α, does not chnge the slopes of the smptotes of memer of the fmil + =. α 33. Sttement : The slope of the common tngent etween the hperol = nd + = m e or. Sttement : The locus of the point of inteeersection of lines = m nd + = is m hperol (where m is vrile nd 0). 33. Sttement : The eqution + + λ = 0 cn never represent hperol. Sttement : The generl eqution of second degree represents hperol if h > Sttement If point (, ) lies in the region II of =, shown in the figure, then < 0 Y = I III II II I = Sttement If (P(, ) lies outside the hperol =, then < 334. Sttement Eqution of tngents to the hperol 3 = 6 which is prllel to the line = is = 3 5 nd = Sttement = m + c is tngent to / / = if c = m Sttement : There cn e infinite points from where we cn drw two mutull perpendiculr tngents on to the hperol = 9 6 III X 85 of 9

5 DOWNLOAD FREE FROM PH.: , Sttement : The director circle in cse of hperol = will not eist ecuse < nd director circle is = Sttement : The verge point of ll the four intersection points of the rectngulr hperol = nd circle + = 4 is origin (0, 0). Sttement : If rectngulr hperol nd circle intersect t four points, the verge point of ll the points of intersection is the mid point of line-joining the two centres Sttement : No tngent cn e drwn to the hperol =, which hve slopes greter thn Sttement : Line = m + c is tngent to hperol =. If c = m 338. Sttement : Eccentricit of hperol 3 3 = 0 is 4/3 Sttement : Rectngulr hperol hs perpendiculr smptotes nd eccentricit = 339. Sttement : The eqution + + λ = 0 cn never represent hperol Sttement : The generl eqution of second degree represent hperol it h > Sttement : The comined eqution of oth the es of the hperol = c is = 0. Sttement : Comined eqution of es of hperol is the comined eqution of ngle isectors of the smptotes of the hperol. 34. Sttement : The point (7, 3) lies inside the hperol 9 4 = 36 where s the point (, 7) lies outside this. Sttement : The point (, ) lies outside, on or inside the hperol = ccording s < or = or > Sttement : The eqution of the chord of contct of tngents drwn from the point (, ) to the hperol 6 9 = 44 is = 44. Sttement : Pir of tngents drwn from (, ) to = is SS = T S = = S = 343. Sttement : If PQ nd RS re two perpendiculr chords of = e, nd C e the centre of hperol = c. Then product of slopes of CP, CQ, CR nd CS is equl to. Sttement : Eqution of lrgest circle with centre (, 0) nd ling inside the ellipse is = 0. Answer 86. C 87. B 88. A 89. C 90. B 9. B 9. B 93. C 94. A 95. A 96. B 97. A 98. C 99. B 300. A 30. A 30. B 303. C 304. B 305. A 306. A 307. A 308. A 309. A 30. B 3. C 3. B 33. A 34. A 35. A 36. C 37. A 38. B 39. A 30. D 3. A 3. A 33. A 34. C 35. C 36. A 37. A 38. D 39. A 330. A 33. B 33. D 333. D 334. C 335. D 336. A 337. A 338. D 339. A 340. A 34. A 34. B 343. B Solution 86. Option (C) is correct. = m + m 0 = 4m. 9 / 4 m 6m 40m + 9 = 0 m =, m 9 = Ever m =,m = Ever prol is smmetric out its is Option (B) is correct An norml to = 4 is 86 of 9

6 DOWNLOAD FREE FROM PH.: , Y + t = t + t 3 If this psses through (λ, λ + ), we get λ + + λ = t + t 3 t 3 + t( - λ) - λ - = 0 = f(t) (s) If λ <, then f (t) = 3t + ( - λ) > 0 f(t) = 0 will hve onl one rel root. So A is true. Sttement is lso true coz (λ + ) > 4λ is true λ. The sttement is true ut does not follow true sttement For the prol =, eqution of norml with slope - is = (-) -3 (-) 3 + = 9, k = 9 Ans. (A). 89. SP = + t = ( + t ) ( + t SQ = + /t ) = = t ( + t ) + = = SP SQ ( + t ),, re in A.P. SP SQ is H.M. etween SP & SQ Hence + = = k = /- = k k 90. = 4 eqution of tngent of slope m = m + m Ans. (C) If it touches = 4 then = 4 (m + /m) 4 4m - = 0 will hve equl roots m D = m + 0 m = m 3 = - m = - So = = 0 (, -) & (-, ) lies on it B is correct. 9. ( ) ( t ) + ( t) = 0 Solve with = 0 t + ( t) = 0 t + t = 0 If it touches -is then ove qudrtic must hve equl roots. SO, D = 0 4 t 4 t = 0 which is correct. B is correct. S(, 0) (t, t) 96. (B) An norml to the prol = 4 is + t = t + t 3 It this psses through (λ, λ + ) t 3 + t( - λ) - λ - = 0 = f(t) s) λ < thn f (t) = 3t + ( - λ) > 0 f(t) = 0 will hve onl one rel root A is true The sttement- is lso true since (λ+ ) > 4λ is true for ll λ. The sttement- is true ut does not follow true sttement-. 87 of 9

7 DOWNLOAD FREE FROM PH.: , = m + m 0 = 4m + 9 / 4 m 6m 40m + 9 = 0 m = /4, m = 9/4 Ever prol is smmetric out its is. 98. (C) Sttement- is true ut Sttement- is flse. 99. (B) If the point (sin α, cos α) lies inside or on the prol + = 0 then cos α + sin α 0 sin α( sin α ) 0 sin α 0, or sin α (A) = ( + ) + is of the form = m( + ) + /m where m =. Hence the line touches the prol. 30. An norml to the prol = 4 is + t = t + t 3 If this psses through (λ, λ + ). We get λ + + λ t = t + t 3. t 3 + t ( - λ) (λ + ) = 0 = f(t) (let) if λ <, then, f (t) = 3t + ( - λ) > 0 f(t) = 0 will hve onl one rel root. sttement I is true. Sttement II is lso true since (λ + ) > 4λ is true for ll λ R ~ {}. Sttement I is true ut does ot follow true sttement II. Hence () is the correct nswer (B) Becuse the common tngent hs to e perpendiculr to =. Its slope is Ellipse is + = focus ( 5,0), e = 3, An point n ellipse ( 3, eqution of circle s the dimeter, joining the points ( 3/, / ) nd focus ( 5,0) is ( 5 ) ( 3) + (. ) = 0 (A) is the correct option () (λ, 3) should stisf the eqution + = 3 λ = ± (A) Here = 4, = 3 nd m = eqution of the tngent is = ± = ± Sttement I is true s it is known fct nd sttement II is oviousl true. However sttement II is not true resoning for sttement I, s coordinte sstem hs nothing to do with sttement I. 3. Given ellipse is + = = 64; 3 = 00 e = ( < ) 5 Now, focl distnce of (, ) on ellipse will e 7 nd 3. Now, for ellipse 7 6, 9, e = = = =. Focus is (e, 0) or ( ) Now rdius of the circle = Distnce etween ( 7, 0 ) nd (0, 3) = 4. 7, of 9

8 DOWNLOAD FREE FROM PH.: , Hence (c) is the correct nswer. 33. Option (A) is correct Sum of the distnce etween foci = e Sum of the focl distnces = e e < e coz e <. Both re true nd it is correct reson. 34. Option (A) is true. Let = m e n chord through (0, 0). This will meet conic t points whose -coordintes re given + m + m = ( + m + m ) = = 0 = 0 Also = m, = m + = m ( + ) = 0 + = 0 mid-point of chord is (0, 0) m. 35. Eqution of PQ (i.e., chord of contct) to the ellipse + = 6 h + k =... () 6 3 An tngent to the ellipse + 4 = 4 is i.e., / cosθ + sinθ =... () () & () represent the sme line h = 3cosθ, k = 3sinθ Locus of R (h, k) is + = 9 Ans. (A) 36. /5 + (-3) /9 = Ends of the mjor is re (0, 6) nd (0, 0) Eqution of tngent t (0, 6) nd (0, 0) is = 6, nd = 0 Anc. (C) = will hve director circle + = = 5 nd we know tht the locus of the point of intersection of two mutull perpendiculr tngents drwn to n stndrd ellipse is its director circle. is correct. 38. B formul p p = = 3. lso foot of perpendiculr lies on uilir circle of the ellipse. B is correct. 3. Sum of distnces etween foci = e sum of the focl distnces = /e e < /e since e <. (A) 3. The sttement- is flse. Since this will represent hperol if h > λ > λ > 4 Thus reson R eing stndrd result is true. (A) 33. () Both Sttement- nd Sttement- re True nd Sttement- is the correct eplntion of Sttement (C) Required tngent is 3 = or = of 9

9 DOWNLOAD FREE FROM PH.: , (C) re of PS S = e sin θ clerl its mimum vlue is e. P( cos θ, sin θ) S ( e, 0) S (e, 0) 37. Tngents cnnot e drwn from one rnch of hperol to the other rnch. Ans. (A) 38. (d) ( λ 3) = λ = 0 or (A) Hperol is ( 4 ) ( 3 ) e = =. 6 4 = Both sttements re true nd sttement II is the correct resoning for sttement I, s for n memer, semi trnsverse nd semi conjugte es re α nd α Hence () is the correct nswer respectivel nd hence smptoters re lws = ±. 33. If = m + c e the common tngent, then c = m... (i) nd c = - m +... (ii) on eliminting c, we get m = m = ±. Now for sttement II, On eliminting m, we get =, Which is hperol. Hence () is the correct nswer. 33. Option (D) is correct. The sttement- is flse coz this will represent hperol if h > λ > λ > 4 The sttmenet-, eing stndrd result, is true The sttement- is flse coz points in region II lie elow the line = / The region- is true (stndrd result). Indeed for points in region II 0 < <. > / / = if c = m c = 3.3 = 5 c = ± 5 90 of 9

10 DOWNLOAD FREE FROM PH.: , rel tngents re = Ans (C) 335. The locus of point of intersection of two mutull perpendiculr tngents drwn on to hperol 336. circle whose eqution is + =. For So director circle does not eist. So d is correct = = 0 =, + = So (0, 0) is verge point which is lso the mid point of line joining the centres of circle & rectngulr hperol is correct The sttement- is flse. Since this will represent hperol if h > λ > λ > 4 Thus reson R eing stndrd result is true. (A) 340. () Both Sttement- nd Sttement- re True nd Sttement- is the correct eplntion of Sttement-. = is its director 34. (A) nd 7 ( 3) > < (B) Required chord of contct is = 44 otined from =. for 39 Yrs. Que. of IIT-JEE & 5 Yrs. Que. of AIEEE we hve distriuted lred ook 9 of 9

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).