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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

గమనిక : ప్రతిభ.ఈనాడు.నెట్లో వచ్చే ప్రకటనలు అనేక దేశాల నుండి, వ్యాపారస్తులు లేదా వ్యక్తుల నుండి వివిధ పద్ధతులలో సేకరించబడతాయి. ఆయా ప్రకటనకర్తల ఉత్పత్తులు లేదా సేవల గురించి ఈనాడు యాజమాన్యానికీ, ఉద్యోగస్తులకూ ఎటువంటి అవగాహనా ఉండదు. కొన్ని ప్రకటనలు పాఠకుల అభిరుచిననుసరించి కృత్రిమ మేధస్సు సాంకేతికతతో పంపబడతాయి. ఏ ప్రకటనని అయినా పాఠకులు తగినంత జాగ్రత్త వహించి, ఉత్పత్తులు లేదా సేవల గురించి తగిన విచారణ చేసి, తగిన జాగ్రత్తలు తీసుకొని కొనుగోలు చేయాలి. ఉత్పత్తులు / సేవలపై ఈనాడు యాజమాన్యానికి ఎటువంటి నియంత్రణ ఉండదు. కనుక ఉత్పత్తులు లేదా సేవల నాణ్యత లేదా లోపాల విషయంలో ఈనాడు యాజమాన్యం ఎటువంటి బాధ్యత వహించదు. ఈ విషయంలో ఎటువంటి ఉత్తర ప్రత్యుత్తరాలకీ తావు లేదు. ఫిర్యాదులు తీసుకోబడవు.

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