ws st circle

8/13/2019 Ws St Circle

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Workshop-st line+pair of st line+circle. By abhijit kumar jha

http://akj259.wordpress.com 1

SECTION - I : (Single Correct Choice Type)

This Section contains 25 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLYONE is correct.

1. The locus of the centre of conic tx2+ 4xy 23 3

2 4 0y tx yt t

is :

(A) 14x 3y 3 = 0 (B) 14xy 3x+ 6y 3 = 0(C) 14y 3x+ 3 = 0 (D) 14xy+ 3x 6y+ 3 = 0

2. If two straight lines represented by the equationy3+ ax y2+ (a2+ b)x2y+ 2x3= 0 are perpendicular to each other, thena, bmay be :(A) a= 2, b= 4 (B) a= 2, b= 4 (C) a= 2, b= 4 (D) None of these

3. Equation ax2+ 2h xy+ by2= 0 represents a pair of lines, combined equation of lines that can be obtained by takingthe mirror image of lines about thex-axis is :(A) ax2+ 2h xy+ by2= 0 (B) bx2+ 2h xy+ ay2= 0 (C) bx2 2hxy+ ay2= 0 (D) by2 2h xy+ ax2= 0

4. A pair of perpendicular lines is drawn through the origin with the line 2x + 3y = 6 an isosceles triangle right angled atorigin. The equation of the line pair is :(A) 5x2 24xy 5y2= 0 (B) 5x2 26xy 5y2= 0 (C) 5x2+ 24xy 5y2= 0 (D) 5x2+ 26xy 5y2= 0

5. Distance between the two lines represented by the line pairx2 4xy+ 4y2+x 2y 6 = 0 is :

(A) 5 (B)1

5 (C) 2 (D) 1

6. A variable line L passing through the pointB(2, 5) intersects the line 2x2 5xy+ 2y2= 0 at P& Qthen the locus ofpointRonLsuch that distanceBP,BR&BQare in harmonic progression.(A) Straight line (B) Cirlce (C) Ellipse (D) Hyperbola

7. The base of triangle passes through a fixed point (f, g) and its sides are respectively bisected at right angle by the linesy2x2= 0 then the locus of its vertex is :

(A)f

y xg

(B)

gy x

f

(C)

fy x

g

(D) None of these

8. Area of the triangle formed by the angle bisectors of the pair of linesx2y2+ 2y 1 = 0 and the linex+y= 3 (in sq.unit) is :(A) 1 (B) 2 (C) 3 (D) 4

9. The equation of pair of lines passing through the origin and having slope mIfor which equation(x 3) (x+ m) + 1 = 0

has integral root is :(A)y2 6xy+ 5x2= 0 (B)y2+ 6xy+ 5x2= 0 (C)y2+ 6xy 5x2= 0 (D)y2 6xy 5x2= 0

10. The linex+y= 1 meets the lines represented by the equationy3 6xy2+ 11x2y 6x3= 0 at the points P, Q, R. If Oisthe origin, then (OP)2+ (OQ)2+ (OR)2is equal to :

(A)85

72 (B)

121

72 (C)

211

72 (D)

217

72

11. Which of the following pairs of straight lines intersect at right angles ?

(A) 2x2=y(x+ 2y) (B) (x+y)2=x(y+ 3x) (C) 2y(x+y) =xy (D)y= 2x

12. If the lines joining the origin to the points of intersection ofy= mx+ 1 withx2+y2= 1 are perpendicular, then :

(A) m= 1 only (B) m= 1 (C) m= 0 (D) none of these

13. If one of lines of pair ax2+ 2h xy+ by2= 0 bisects the angle between positive directions of axis a, b, hsatisfies therelation :

(A) a+ b= 2 | h| (B) a+ b= 2h (C) a b= 2 | h| (D) (a b)2= 4h2


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14. The difference of the tangents of angles which the linesx2(sec2 sin2) 2xytan +y2sin2= 0 makes with thex-axis is kthan the equation of line passing through origin and having slope |k| is :

(A)x 2y= 0 (B)y 2x= 0 (C)y+ 2x= 0 (D)x+ 2y= 0

15. If the lines represented byx2 2pxy y2= 0 are rotated about the origin through an angle , one clockwise directionand other in anticlockwise direction, then the equation of the bisectors of the angle between the lines in new positionis :

(A)px2

+ 2xypy2

= 0 (B)px2

+ 2xy+py2

= 0 (C)x2

2pxy+y2

= 0 (D) None of these

16. The equation of image of pair of linesy2= (x 1)2iny-axis is :

(A)x2+y2+ 2x+ 1 = 0 (B)x2y2+ 2x 1 = 0 (C)x2y2+ 2x+ 1 = 0 (D) none of these

17. The equation 2 2 2 2( 2) ( 2) 4x y x y represents :

(A) circle (B) pair of lines

(C) parabola (D) line segmenty= 0, 2 x2

18. One of the bisectors of the angle between the lines a(x 1)2+ 24(x 1)(y 2) + b(y 2)2= 0 isx+ 2y 5 = 0 thenthe other bisector is :

(A) 2xy= 0 (B) 2xy1 = 0 (C) 2xy 2 = 0 (D) None of these

19. The linex+y= 1 meets the lines represented by the equation y3 xy2 14x2y+ 24x3= 0 at the pointsA, B, Cif Oisthe origin, then OA2+ OB2+ OC2is :

(A)22

9 (B)

85

72 (C)

181

72 (D)

221

72

20. If the straight lines joining the origin to the points of intersection of the two curvespx2+ 2qxy+ ry2+ 2sx= 0 andp'x2+ 2q'xy+ r'y2+ 2s'x= 0 are at right angles then s'(p+ r) is equal to :

(A) s(p' + r') (B) s(q+ r) (C) s(r+ s) (D) none of these

21. If (x, y) ax2+ 2h xy+ b y2+ 2gx+ 2fy+ c= 0 represents a pair of straight lines, then the equation of any linepassing through the point of intersection of the given lines is :

(A) gxfy= c (B) gx+fy= c (C) gx+fy= c (D) gx+fy+ c= 0

22. If the product of perpendiculars drawn from the point (1, 1) on the lines ax2+ 2hxy+ by2= 0 is 1, then :

(A) h(a b) + ab= 0 (B) h(a+ b) ab= 0 (C) h(a+ b) + ab= 0 (D) h(a b) ab= 0

23. All chords of the curve 3x2y2 2x+ 4y= 0, which subtend a right angle at the origin pass through a fixed point.The square of its distance from the origin is :

(A) 2 (B) 3 (C) 4 (D) 5

24. The straight linex2+ 4xy+y2= 0 and the linexy+ 4 = 0 forms a :

(A) equilateral triangle (B) right angled triangle

(C) obtuse angled triangle (D) none of these

25. If pairs of straight linesx2 2pxyy2= 0 andx2 2q xy y2= 0 be such that each pair bisects the angle between theother pair, then p.q is :

(A) 2 (B) 1 (C) 1 (D) 2


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SECTION - II : (Multiple Correct Choice Type)

This Section contains 10 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONEOR MORE may be correct.

26. Type of quadrilateral formed by two pair of lines 6x2 5xy 6y2= 0 and 6x2 5xy 6y2+ 5y+x 1 = 0 is :

(A) square (B) rhombus (C) parallelogram (D) rectangle

27. Two pairs of straight lines have the equationsy2+xy 12x2= 0 and ax2+ 2h xy+ by2= 0. One line will be commonamong them if :

(A) a= 3(2h+ 3b) (B) a= 8 (h 2b) (C) a= 2(b+ h) (D) a= 3 (b+ h)

28. The combined equation of three sides of a triangle is (x2y2) (2x+ 3y 6) = 0. If (2, a) is an interior point end (b, 1)is an exterior point of triangle, then

(A)10

23

a (B)10

23

a (C)9

12

b (D) 1 < b< 1

29. If the two lines represented byx2(tan2+ cos2) 2xytan +y2sin2= 0 make angles withx-axis, then

(A) tan + tan = 4 cosec 2 (B) tan + tan = sec2+ tan2

(C) tan tan 2 (D)tan 2 sin 2

tan 2 sin 2

30. Equation of pair of lines passing through (1, 1) and parallel to lines 2x2+ 5xy+ 3y2= 0 is :

(A) 2(x 1)2+ 5(x 1) (y+ 1) + 3(y+ 1)2= 0 (B) 3(x 1)2 5(x 1) (y+ 1) + 2(y+ 1)2= 0

(C) 2x2+ 5xy+ 3y2+x+y= 0 (D) 3x2 5xy+ 2y2 11x+ 9y+ 10 = 0

31. If one of the lines given by the equation 2x2+pxy+ 3y2= 0. Coincide with one of those given by 2x2+ qxy 3y2= 0and the other lines represented by them are perpendicular, then

(A)p= 5 (B)p= 5 (C) q= 1 (D) q= 1

32. If slope of one of the lines represented by ax2 6xy+y2= 0 is square of the other then :

(A) a= 1 (B) a= 27 (C) a= 4 (D) a= 8

33. There is a pair of points, one on each of the lines whose combined equation is (4x 3y+ 5) (6x+ 8y+ 5) = 0. If theyare such that the distance of the point on one line is 2 units from the other line then the points one :

(A)1 9 1

, & , 110 5 2

(B)1 23 7

, 1 & , 2 10 3

(C)1 9 23 7

, & , 10 5 10 5

(D) None of these

34. Two pairs of straight lines have the equationsy2+xy 12x2= 0 and ax2+ 2hxy+ by2= 0 one line will be common

among them if.(A) a= 3(2h+ 3b) (B) a= 8 (h 2b) (C) a= 2(b+ h) (D) a= 3 (b+ h)

35. Ifx2+ 2hxy+ y2= 0 represents the equations of straight lines through the origin which makes angle with the linex +y= 0 then :

(A)1

cos2h

(B)1

cos2

h

h

(C)

12sin

2

h

h

(D)

1tan

1

h

h


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SECTION - III : (Paragraph Type)

This Section contains 3 paragraphs. Based upon the first paragraph 2 multiple choice questions and based upon the secondparagraph 3 multiple choice questions have to be answered. Each of these questions has four choice (A), (B), (C) and (D) outof WHICH ONLY ONE CORRECT.

Paragraph for Question Number 36 to 38

In an acute angled triangleABC, A(0, 0) and joint equation of altitude throughBand Cis

2x

2

+ 5xy+ 2y

2

9x 9y+ 9 = 036. Slope of line BC is equal to :

(A) 1 (B) 1 (C) 4/3 (D) 3/4

37. Circum-radius of triangle ABC is equal to :

(A)7 2

8 (B)

3 2

8 (C)

5 2

8 (D)

11 2

8

38. Joint equation of lineABandACis :(A) 2x2 5xy+ 2y2= 0 (B)x2 5xy+y2= 0 (C) 3x2xyy2= 0 (D) 3x2+xyy2= 0

Paragraph for Question Number 39 to 41

Secant along line y = mx + c intersect parabola y2 = 4ax at A & B. Such that AOB = 90 (where O (0, 0)(also a> 0).

39. Secant line will always pass through a fix point P whose co-ordinate is :(A) (a, 0) (B) (2a, 0) (C) (4a, 0) (D) (8a, 0)

40. Locus of mid-point of chord AB is a parabola whose latus rectum is equal to :(A) 4a (B) 2a (C) a (D) 8a

41. Least circum-radius of such a triangle AOB is equal to :

(A) 6a (B) 8a (C) 4 2a (D) None of these

Paragraph for Question Numbers 42 to 44

Joint equation of two lines is 2x2 5xy+ 2y2= 0 and P(1, 1) is a point.

42. From P, perpendiculars PAand PBare drawn to both lines, (whereA&Bare the foot of perpendiculars), then lengthofABis equal to

(A) 10 2 (B)3 2

5 (C)

7 2

3 (D) 9 2

43. IfL1= 0 & L2 = 0 are two lines passing through P such that chords intersected by them between the given lines isdivided at Pin the ratio 2 : 1 then joint equation ofL1= 0 andL2= 0 is :(A) 2x2y2+ 2xy= 0 (B)xyxy+ 1 = 0 (C) 2xy 2yx+ 1 = 0 (D) None of these

44. If line joining pointAon one branch of the given line and pointBon the other branch of the given line is bisected atpoint P then equation ofABis :(A) 2x+y 2 = 0 (B)x+ 2y 2 = 0 (C) 4x+ 7y 11 = 0 (D) None of these

SECTION - IV : (Assertion & Reason Type)

This section contains 3 Assertion-Reason type questions. Each question contains STATEMENT-1 (Assertion) and

STATEMENT-2 (Reason). Each question has 4 choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

In each of the following questions, two statements are given. Choose the correct code (A), (B), (C) or (D) definedbelow :(A) Statement-1 and Statement-2 are true, Statement-2 is the correct explanation of Statement-1.(B) Statement-1 and statementr-2 are true but Statement-2 is not the correct explanation of Statement-2 is false.(C) Statement-1 is true and Statement-2 is false.(D) Statement-1 is false and Statement-2 is true.

45. Statement-1 : If 2h= a+ b, then one line of the pair of lines ax2+ 2h xy+ by2= 0 bisects the angle between co-ordinate axes in positive quadrant.

Statement-2 : If ax+ y (2h+ a) = 0 is a factor of ax2+ 2h xy+ by2= 0 then a+ b+ 2h= 0.


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46. Statement-1 : The combined equation ofL1,L2is 2x2+ 6xy+y2= 0 and that ofL1',L2' isx

2 6xy+ 2y2= 0. If anglebetweenL1,L2' is , then angle betweenL2,L1' is also .

Statement-2 : If the pair of line L1L2= 0,L1'L2' = 0 are equally inclined then angle between L1L2' = angle betweenL2L1'.

47. Statement-1 : If algebric sum of the perpendicular drawn from points (2, k1), (3, k2), (7, k3), (h1, 4), (h2, 5), (h3, 3)to a family of line passing through (2, 1) is zero, then the value of non-negative real numbers h1, h2, h3,k1, k2, k3are all zero.

Statement-2 : If algebric sum of the perpendicular from n points to a variable line is zero. Then the variable linealways passes through the mean of the given n points.

SECTION - V : (Matrix Type)

This Section contains 3 questions. Each question has four statements (P, Q, R and S) given in Column I and four statements(A, B, C and D) in Column II. Any given statement in Column I can have correct matching with one or more statement(s)given in Column II. For example, if for a given question, statement Q matches with the statements given in B and C, then forthat particular question, against statement Q, darken the bubbles corresponding to B and C in the OMR.

48. Given equation represents two coincident linesx2+ 9y2+pxy+ qx+ ry+ 1 = 0, then :

Column I Column II

(A) pequals to (P) 6(B) qequals to (Q) 2

(C) requals to (R) 3

(D) . . .

12

p q ris equal to

(S) 72

49. Consider second degree equation ax2+ 2h xy+ by2+ 2gx+ 2fy+ c= 0 represents pair of lines then :Column I Column II

(A) If (x1, y1) is the point of intersection of the twolines, then (ax1+ hy1) (hx1+ by1) =

(P)

2 2( ) 4

c

a b h

(B) af2+ bg2+ ch2= (Q) ab

(C) The lines are parallel if h2= (R) gf

(D) Product of perpendicular from the origin is (S) abc+ 2fgh

50. Distance between the linesColumn I Column II

(A) 4x2+ 12xy+ 9y2 16x 24y+ 7 = 0(P) 6

5

(B) 9x2+ 24xy+ 16y2+ 12x+ 16y 5 = 0(Q) 6

13

(C) x2+ 2xy+y2+ 4x+ 4y 21 = 0(R) 10

2

(D) 9x2 12xy+ 4y2+ 6x 4y 3 = 0(S)

413

SECTION - VI : (Integer Type)

This Section contains 10 questions. The answer to each question is a single digit integer, ranging from 0 to 9. The correctdigit below the question number in the OMR is to be bubbled.


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51. If the angle between the pair of straight lines y2 sin2 xy sin2 + x2 (cos2 1) = 0 is , then the value of3sin 32sin 9

6

must be ____ .

52. If two of the lines represented byx4+x3y+ cx2y2xy3+y4= 0 bisects the angle between the other two then the valueof | c| is ____ .

53. Suppose the distance between two points P(x1, y1) and Q(x2,y2) defined as d(P, Q) = man {|x2x1|, | y2y1|}, ofA(1, 2), d(P, A) = 4 and if area of the region bounded by locus of point Pis k2, then kis ____ .

54. A line passes through the point P(2, 3) and makes an angle of 30 with the positive direction of x-axis. It meets thelines represented byx2 2xy y2= 0 at the points A and B such that PA . PB= aand if kis the sum of digits of thevalue of 7(a 17)2then k/7 is ____ .

55. If ax2 2h xy ay2= 0 represent a pair of straight line forming with 2x+ 3y= 6 an isosceles right angled triangle atthe origin then | h a| is ____ .

56. If two of the lines given by 3x3+ 3x2y 3xy2+ dy3= 0 are at right angles then absolute value of their slope is ____ .

57. If two of the lines a x3+ bx2y+ c xy2+ d y3= 0 (a0) make complementary angles with x-axis in anticlockwisesense then the value of a (a c) + d(b d) is ____ .

58. If the lines represented by 2x25xy+ 2y2= 0 be the two sides of a parallelogram and the line 5x + 2y = 1 be one of itsdiagonal and if area of the parallelogram is kthen 36 kis _____ .

59. If the linesx2+ 4xy+y2= 0 and the linexy= 4 form a triangle of side length a, b, cthen2 2 2

16

a b c is ____ .

60. The base of a triangle passes through a fixed point (8, 0) and its sides are respectively bisected at right angles by thelines y2 8xy 9x2= 0. If the locus of it vertex is a conic then absolute value of sum of xandyco-ordinate of itscentre is ____ .

* * * * *

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