1. The equation of a straight line which cuts off an intercept of '5' units on the negative direction of Y - axis and makes an angle of 120º with the positive direction of X - axis is -
A: x + y + 5 = 0
2. The equation of a straight line cutting of an intercept -1 from Y - axis and being equally inclined to the axes is....
A: y = ± x − 1
3. The equation of a line that has 'y' intercept '4' and is perpendicular to the line joining (2, −3) and (4, 2) is....
A: 2x + 5y − 20 = 0
4. The equation of the perpendicular bisector of the line segment joining the points A (2, 3) and B (6, −5) is ....
A: x − 2y − 6 = 0
5. The equations of the medians of the triangle ABC whose vertices are A (2, 5) B (−4, 9) C (−2, −1) are ....
A: x − 5y + 23 = 0
7x + 4y − 8 = 0
8x − y + 15 = 0
6. In what ratio does the line joining the points (2, 3) and (4, 1) divide the segment joining the points (1, 2) and (4, 3)?
A: 1 : 1
7. If A (7, -1); B (-2, 8); C (1, 2) are the vertices of triangle ABC then which of the following is not an equation of the altitude?
A: x + y + 1 = 0
8. Equations of the diagonals of square formed by the lines x = 0, y = 0 and x = 1; y = 1 are
A: x − y = 0; x + y = 1
9. Two consecutive sides of a parallelogram are 4x + 5y = 0 and 7x + 2y = 0. If the equation to one diagonal is 11x + 7y = 9, then the equation of the other diagonal is
A: y − x = 0
10. A line 4x + y = 1 through the point A (2,−7) meets the line BC whose equation is 3x + 4y + 1 = 0 at the point B. Find the equation of the line AC so that AB = AC.
A: 52x + 89y + 519 = 0
11. The intercept of a line between positive axes is divided by the point (2, 3) internally in the ratio 1 : 2 then cartesian equation in normal form is ....
12. The line 3x - 4y + 7 = 0 is rotated through the angle in clock wise direction about the point (−1, 1) the equation of the line in its new position is -
A: 3x + 4y = 1
13. If the straight lines ax + by + c =0 and xcosα + ysinα = c enclose an angle between them and meet the straight line xsinα − ycosα = 0 in the same point then the value of a2 + b2 =
14. The equations of the line which passes through the point (3, 4) and the sum of intercepts on the axes is 14
A: x + y = 7; 4x + 3y = 24
15. The equation of the straight line that makes equal intercepts on the axes and passes through the point (2, 3) is .....
A: x + y = 5
16. The equation of the straight line that passes through the point (−5, 4) and is such that the portion intercepted between the axes is divided by the point in the ratio 1 : 2 is .....
A: 8x − 5y = 60
17. A variable straight line is drawn through the point of intersection of straight lines and and meets the coordinate axes at 'A' and 'B' if the locus of mid point AB is kxy (a + b) = ab(x + y) then k = ......
18. If the line moves in such a way that where 'c' is constant. Then the locus of the foot of the perpendicular from the origin is
A: x2 + y2 = c2
19. The equation of straight line which passes through the point (−3, 3) and cuts off +ve intercepts on the axes whose sum is 7
A: 4x + 3y = 12
20. The equation of a straight line, the portion of which intercepted between the axes is divided by the point (−5, 4) in the ratio 1 : 2
A: 5y − 8x = 60
21. The sum of intercepts (on co-ordinate axes) made by a straight line joinng the point (a, b) to the point of intersection of the lines and is
22. The equation of line passing through the point (2, 3) and making an intercept of length '2' between the lines y + 2x = 3 and y + 2x = 5
Ans: 3x + 4y = 18
23. The number of integral values of 'm' for which the abscissa of the point of intersection of the lines 3x + 4y = 9 and y = mx + 1 is an integer is-
24. The area of ∆le formed by the line 2x + y = a with two mutually perpendicular lines in a plane forming an isosceles ∆le which are drawn from (0, 0) is .... sq. units
25. The area enclosed by is ...... sq. units
26. A square of side 'a' lies above the X − axis and has one vertex at the origin. The side passing through the origin makes an angle with the positive direction of X - axis then the equation of its diagonals:
Ans: x = 0; y = a
27. A square of side 'a' lies above the X - axis and has one vertex at the origin the side passing through the origin makes an angle α (0 < α < ) with the positive direction of X - axis then the equation of its diagonal not passing through origin is...
Ans: y (cosα + sinα) + x (cosα − sinα) = a
28. The point A (2, 1) is translated parallel to the line x - y = 3 by a distance '4' units. If the new position A' is in third quadrant then the coordinates of A' are
Ans: (2−2, 1−2)
29. The line meets the Y- axis; X−axis at Aand B respectively. A square ABCD constructed on the line segment AB away from the origin. The coordinates of the vertex of square farthest from the origin is ....
Ans: (4, 7)
30. One of the diagonals of the square is the portion of the line intercepted between the axes then the extremities of the other diagonal are ...
Ans: (5, 5) (−1, 1)
31. The Coordinates of two consecutive vertices A and B of regular hexagon ABCDEF are (1, 0) and (2, 0) respectively. The equation of the diagonal CE is ...
Ans: x + y = 4
32. A variable straight line is drawn through a given point 'O' to cut two fixed straight lines in 'R' and 'S' on, it is taken point 'P' such that then the locus of the point 'P' is:
Ans: A Straight line
33. A line through A(−5, −4) meets the line x + 3y + 2 = 0 and 2x + y + 4 = 0 and x − y − 5 = 0 at the points B, C and D respectively. If Then the equation of the line is
Ans: 2x + 3y + 22 = 0
34. The ends of the diagonal of a square are (2, -3) and (-1, 1) then the other vertices are
35. A line is such that it's segment between the straight lines 5x − y − 4 = 0 and 3x + 4y − 4 = 0 is bisected at the point (1, 5) then its equation is
Ans: 83x − 35y + 92 = 0
36. The equations of the straight lines passing through (-2, -7) and having an intercept of length '3' between the straight lines 4x + 3y − 12 =0 and 4x + 3y − 3 = 0 is
Ans: x + 2 = 0; 7x + 24y + 182 = 0
37. If p and q are the distances of origin from the lines x secα + y cosecα = k and
x cosα − y sinα = k cos 2α then which of the following is true?
a) 4p2 + q2 = 2k2 b) 4p2 + q2 = k2
c) p2 + 4q2 = k2 d) 4p2 - q2 = k2
Ans: 4p2 + q2 = k2
38. The area of the parallelogram formed by the lines a1x + b1y + c1 = 0; a2x + b2y + c2 = 0 and a1x + b1y + d1 = 0; a2x + b2y + d2 = 0 is
39. The length of the perpendicular from the origin to a line is '7' and the line makes an angle of 150º with positive direction of Y − axis. Then the equation of the line is
Ans:x + y − 14 = 0
40. A line forms a triangle of area 54 sq. units with the coordinate axes then the equation of the line if the perpendicular drawn from the origin to the line makes an angle of 60º with X − axis is .....
Ans: x + y−18 = 0
41. A straight canal is miles from a place and the shortest route from this place to the canal is exactly North−East. A village 3 miles north and 4 miles east from the place. Does it lie by the nearest edge of the canal?
42. A rectangle has two opposite vertices at the points (1, 2) and (5, 5) if the other vertices lie on the line x = 3. Then the other vertices of the rectangle are-
Ans: (3, 1) (3, 6)
43. If the equation of base of an equilateral triangle is 2x−y = 1 and the vertex is (−1, 2) then the length of the side of the triangle is
44. If the point (4, 7) and (cosθ, sinθ) where (0 < θ < Π) lie on the same side of the line x + y − 1 = 0 then 'θ' lies in-
Ans: Ist Quadrant
45. If the point (O, b) lies on or inside the triangle having the sides 3x + y + 2 = 0; 2x − 3y + 5 = 0 and x + 4y − 14 = 0 then b lies in -
46. If the point (α, α2) lies inside the triangle formed by the lines 2x + 3y − 1 = 0; x + 2y − 3 = 0 and 5x − 6y − 1 = 0 then α lies in -
47. If the point (Sin2θ, Sinθ) lies inside the square formed by the lines xy = 0 and 4xy − 2x − 2y + 1 = 0 then all values of θ lies in
48. If the point (a2, a) lies in the region corresponding to the acute angle between the lines 2y = x and 4y = x then values of 'a' belongs to-
Ans: (2, 4)
49. The range of α if (α, α2) lies inside the triangle having sides along the lines 2x + 3y = 1 and x + 2y = 3 and 6y = 5x − 1 is -
50. If the lines (a − b − c)x + 2ay + 2a = 0 and 2bx + (b − c − a)y + 2b = 0 and (2c + 1)x + 2cy + (c − a − b) = 0 are concurrent and if a + b + c ≠ 0 then (a + b + c)2 =
51. If the lines ax + y + 1 = 0; x + by + 1 = 0 and x + y + c = 0 are concurrent (a ≠ b ≠ c ≠ 1), then the value of
52. The set of lines ax + by + c = 0 where 3a + 2b + 4c = 0 is concurrent at the point -
53. If a, b, c are in A.P. then the straight lines ax + by + c = 0 will always pass through the point:
Ans: (1, −2)
54. The value of λ such that the straight line (2x + 3y + 4) + λ (6x − y + 12) = 0 is parallel to Y − axis is ....
55. The lines 2x + 3y − 8 = 0; 5x − 6y + 7 = 0 and px + qy = 1 are concurrent then the line x + 2y − 1 = 0 always passes through the point:
Ans: (p, q) or (-p, -q)
56. For what value of k will the following lines be concurrent?
3x − 4y + 5 = 0; 7x − 8y + 5 = 0; 4x + 5y + k = 0
57. The equations to the straight lines passing through the point (2, 3) and equally inclined to the lines 3x − 4y − 7 = 0 and 12x − 5y + 6 = 0 are...
Ans: 7x + 9y = 73; 9x = 7y + 1
58. A ray of light is sent along the line x − 2y − 3 = 0 upon reaching the line 3x − 2y − 5 = 0 the ray is reflected from it. The equation of line containing the reflected ray is -
Ans: 29x − 2y − 31 = 0
59. The bisector of the angle between the lines 2x + y−6 = 0 and 2x − 4y + 7 = 0 which contains the point (1, 2). is
Ans: 6x − 2y − 5 = 0
60. The equation of the bisector of the acute angle between the lines 12x + 5y − 2 = 0 and 3x - 4y + 7 = 0. is
Ans: 11x − 3y + 9 = 0
(61 - 63): Passage
A straight line L with negative slope passes through the point (9, 4) and cuts the positive coordinate axes at the points P and Q respectively.
61. Minimum Value OP + OQ as L varies (O is origin):
62. Area of ∆OPQ when OP + OQ becomes minimum is -
Ans: 75 sq. units
63. Let R be a moving point on the xy plane such that OPRQ becomes rectangle then locus of 'R' as L varies is -
64. Matrix Matching
Consider the triangle formed by the lines y + 3x + 2 = 0; 3y − 2x − 5 = 0 and x + 4y − 14 = 0
Ans: (i) s; (ii) r; (iii) q; (iv) p.