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PAIR OF STRAIGHT LINES

1. If the slope of one of lines ax2 + 2hxy + by2 = 0 is thrice that the other, then  is
A: 

 

2. If  the lines ax2 + 2hxy + by2 = 0 are equally inclined with coordinate axes, then
A: h = 0, ab < 0

 

3. The equation of the pair of lines passing through (1, 2) and parallel to the coordinate axes is
A: xy - 2x - y + 2 = 0


4. x2 + k1y2 + 2k2y - a2 = 0 represents a pair of perpendicular lines if
A: k1 = -1; k2 = -a

 

5. The lines joining the origin to the points of intersection of x2 + y2 + 2gx + c = 0 and x2 + y2 + 2fy - c = 0 are at right angles is
A: g2 - f2 = 2c

 

6. If a(x - 1)2 + 2h(x - 1)(y - 2) + b(y - 2)2 = 0 has one angular bisector 2x + 3y - 8 = 0, then other bisector is
A: 3x - 2y + 1 = 0

 

7. The distance between the parallel lines 9x2 - 6xy + y2 + 18x - 6y + 8 = 0 is
A: 

 

8. If the slope of one of the lines represented by ax2 + 2hxy + by2 = 0 be the square of the other, then
A: 

 

9. The value of  'h' so that the equation 6x2 + 2hxy + 12y2 + 22x + 31y + 20 = 0 represent two straight lines is
A: 

 

10. The angle between the lines joining the origin to the point of intersection of the line y = 3x + 2 with the curve x2 + 2xy + 3y2 + 4x + 8y = 11 is
A: 

 

11. Distance between the parallel lines (x + 2y)2 + 13(x + 2y) + 180 = 0 is
A: 5

 

12. The triangle formed by 2x2 - 3xy + y2 = 0 and x + y - 1 = 0 is
A: Right angled triangle

 

13. If 4a2 + 9b2 - c2 + 12ab = 0 then the family of straight lines ax + by + c = 0 is concurrent at the point

      A) (2, 3)    B) (-2, -3)    C) (-3, -4 )    D) (3, -4)
A: A, B

 

14. If the pair of lines ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 intersect on Y-axis, then
      A) f2 = bc     B) abc = 2fgh    C) bg2 ≠ ch2    D) 2fgh = bg2 + ch2
A: A, D 

 

15. If the two lines represented by x2(tan2 θ + cos2 θ) - 2xy tan θ + y2 sin2 θ = 0 makes an angles α, β with positive X-axis, then
     A) tan α + tan β = 4 cosec 2θ     B) tan α tan β = sec2 θ + tan2 θ
     C) tan α - tan β = 2                     D) tan α + tan β = 2
A: A, C

 

16. If x2 + 2hxy + y2 = 0 represents the equation of pair of straight lines through origin and which makes an angle 'α' with the line x + y = 0 then, h =
A: sec 2α

 

17. The line x + y = 1 meets the lines represented by the equation y3 - 6xy2 + 11x2y - 6x3 = 0 at the points P, Q, R and if  'O' is the origin, then (OP)2 + (OQ)2 + (OR)2 =
A: 

 

18. 2x2 - 5xy + 2y2 = 0 represents two sides of a triangle whose centroid (2, 3), then area of triangle is
A: 6 sq.units

 

19. The equation of image of pair of lines y =  in Y-axis is
A: 4x2 - y2 + 4x + 1 = 0

 

20. If two of the lines represented by x4 + x3y + cx2y2 - xy3 + y4 = 0 bisect the angle between the other two, then c =
A: -6

 

 

21. Let PQR be a right angled isosceles triangle right angled at P(2, 1). If the equation of the line QR is 2x + y = 3, then the equation representing the pair of lines PQ and PR is
Ans: 3x2 - 3y2 + 8xy - 10x - 15y + 25 = 0

 

22. Two pairs of straight lines have the equations y2 + xy - 12x2 = 0 and ax2 + 2hxy + by2 = 0. A 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)
Ans: A, B

 

23. If the line y = mx is one of the bisector of the lines x2 + 4xy - y2 = 0, then m =
    A)         B)           C)           D) 
Ans: A, C

 

Passage (24 - 26)
If the lines represented by 2x2 - 5xy + 2y2 = 0 be the two sides of parallelogram and the line 5x + 2y = 1 be one of its diagonal, then

 

24. The equation of the other diagonal is
Ans: 11x - 10y = 0

 

25. The centroid of the parallelogram is
Ans: 

 

26. The area of the parallelogram is ....... sq units.
Ans: 

 

27. The equations of the pairs of opposite sides of rectangle are x2 - 7x + 6 = 0 and y2 - 14y + 40 = 0, the equation of the diagonal nearer to the origin is
Ans: 6x - 5y + 14 = 0

 

28. The lines represented by the equation x- y2 - x + 3y - 2 = 0 are
Ans: x - y + 1 = 0, x + y = 2

 

29. The length of the side of the square formed by the lines 2x2 + 3xy - 2y2 = 0, 2x+ 3xy - 2y2 + 3x + y + 1 = 0 are
Ans: 

 

30. The circumcentre of the triangle formed by the lines xy + 2x + 2y + 4 = 0 and x + y + 2 = 0 is
Ans: (-1, -1)

 

31. Area enclosed by the pair of lines x2 - y2 - 2x + 1 = 0 and x2 - y2 + 2x + 1 = 0 is
Ans: 2 sq.units

 

32. The quadrilateral formed by the pair of lines x- 5x + 6 = 0, 9x2 + 24xy + 16y2 + 3x + 4y - 6 = 0 is
Ans: Parallelogram

 

33. The four lines y2 - 4y + 3 = 0 and x2 + 4xy + 4y2 - 5x -10y + 4 = 0 forms a
Ans: Parallelogram

 

34. The intercept made by the pair of lines 6x2 - 7xy - 3y2 - 24x - 3y + 18 = 0 on the axis is
Ans: 2

 

35. The product of perpendicular distances from the origin on the pair of straight lines 12x2 + 25xy + 12y2 + 10x + 11y + 2 = 0 is
Ans: 

 

36. If  λx2 + 6xy + 9y2 + 4x + 12y + 3 = 0 represents a pair of straight lines then λ =
Ans: 1

 

37. If the equation x2 + (λ + µ)xy + λµy2 + x + µy = 0 represents two parallel lines, then
Ans: λ = µ

 

38. If the pair of lines 12x2 + 7xy - 12y2 = 0 and 12x2 + 7xy - 12y2 - x + 7y + c = 0 forms a quadrilateral having area    sq units, then c =
Ans: 1

 

39. Area of triangle formed by the lines xy + 2x + 2y + 4 = 0 and x + y + 2 = 0 is
Ans: 2 sq. units

 

40. If the straight line represented by the equation x2 + 6xy + 9y2 + 4x + 12y - 5 = 0 meet the X - axis at A, C and Y-axis at B, D respectively, then the area of trapezium ABCD is (in sq. units)
Ans: 6

 

41. Orthocentre of the triangle formed by the lines x + y + 1 = 0 and 2x2 - xy - y2 + x + 2y - 1 = 0 is
Ans: (-1, 0)
42. If α, β, γ are real roots of the equation x3 - 3px2 + 3qx - 1 = 0, then the centroid of the triangle whose vertices are   is
Ans: (p, q)

 

43. All chord of a curve 3x2 - y2 - 2x + 4y = 0 which subtends a right angle at the origin passes through a fixed point, then it is
Ans: (1, -2)

 

44. The lines joining the origin to the points of intersection of the line x - y = 2 with the curve 5x2 + 12xy - 8y2 + 8x - 4y + 12 = 0 are equally inclined to
Ans: xy = 0

 

45. The curve x2 + y2 + 2gx + 2fy + c = 0 intercepts on the line lx + my = 1 a length which subtends a right angle at the origin, then 
Ans: 

 

Passage (46 - 49)
S ≡ x2 + 4xy + y2 = 0; L = x - y + 1 = 0 are the three sides of a triangle, then answer the following.

 

46. Angle between the pair of lines S ≡ 0 is
Ans: 60°

 

47. Area of the triangle formed by S ≡ 0 and L = 0 is
Ans:    sq.units

 

48. Distance from (1, 2) to pair of lines S ≡ 0 is
Ans:  

 

49. Orthocentre of the triangle formed by S ≡ 0 and L = 0 is
Ans: (-1, -1)

 

MATRIX MATCH
 

50. S = 3x2 + xy - 2y2 = 0 represents a pair of lines, then match the following.

Ans:  i) q  ii) r   iii) p  iv) s

Posted Date : 19-02-2021

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