goyalsmath.comgoyalsmath.com/straight lines.pdf(1) a straight line parallel to x-axis (2) a circle through the origin (3) circle with centre at the origin (4) a straicht line parallel
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Goyal
's M
ath
TOPICS COVERED
STRAIGHT LINES AND PAIR OF STRAIGHT LINES
1. The points A(0, –1), B(2, 1), C(0, 3), D (–2, 1)are the vertices of a(1) square (2)retangle
(3) parallelogram (4) none of these
2. P(3, 1), Q(6, 5) and R(x, y) are three pointssuch that the angle PRQ is a right and the area
of 7,PRQ∆ = then the number of such points
R is(1) 0 (2) 1
(3) 2 (4) 4
3. ABCD is a square ( ) ( )1,2 3, 4 .A B≡ ≡ − If
line CD passes through (3, 8), then mid-point ofCD is(1) (2, 6) (2) (6, 2)
(3) (2, 5) (4)24 1
,5 5
4. The area of the triangle formed by the lines
, 0y ax x y a= + − = and the y - axis is equal to
(1)1
2|1 |a+ (2)
2
|1 |
a
a+
(3)1
2 1
a
a+ (4)
2
2|1 |
a
a+
5. If P (1, 0), Q(–1, 0) and R (2, 0) are three givenpoints, then the locus of points S satisfying the
relation 2 2 22SQ S SP+ = is
(1) a straight line parallel to x-axis
(2) a circle through the origin
(3) circle with centre at the origin
(4) a straicht line parallel to y-axis
6. The straight lines 7 2 10 0x y− + = and
7 2 10 0x y+ − = forms an isosceles triangle
with the line y = 2, area of this triangle is equalto(1) 15/7 sq. units (2) 10/7 sq. units
(3) 18/7 sq. units (4) none of these
7. A line passes through (2, 2) and is perpendicular
to the line 3 3,x y+ = Its y-intercept is
(1)1
3(2)
2
3
(3) 1 (4)4
3
8. If one of the diagonals of a square is along theline x = 2y and one of its vertices is (3, 0), thenits sides through this vertex are given by theequations
(1) 3 9 0,3 3 0y x y x− + = + − =
(2) 3 9 0,3 3 0y x y x+ + = + − =
(3) 3 9 0,3 3 0y x y x− + = − + =
(4) 3 3 0,3 9 0y x y x− + = + + =9. The equation of the diagonal through origin of
the quadrilateral formed by the lines
0, 0, 1 0x y x y= = + − = and 6 3 0x y+ − = is
(1) 4 3 0x y− = (2) 3 2 0x y− =
(3) x y= (4) 0x y+ =10. The equation of the base of an equilateral
triangle is 2x y+ = and the vertex is
( )2, 1− .Length of its side is
(1)1
2(2)
3
2
(3)2
3(4) 2
11. If 2 4 0 2 5 0x y and x y− + = + − = are the
sides of a isoceles triangle having area 10sq.units .Equations of third side is
differential real numbers ,are :(1) collinear (2)vertices of square
(3) vertices of Rhombus
(4) concyclic
21. The incentre of the triangle with the vertices
( ) ( ) ( )1, 3 , 0,0 2,0 :and is
(1)3
1,2
(2)2 1
,3 3
(3)2 3
,3 2
(4)1
1,3
22. A pair of straight lines drawn thrown the origin
form with the line 2 3 6x y+ = an isocelles
right angled triangle ,then the lines and the areaof the triangle thus formed is :
(1)36
5 0,5 0;13
x y x y− = + = ∆ =
(2)12
3 0, 3 0;17
x y x y− = + = ∆ =
(3)13
5 0, 5 0;5
x y x y− = + = ∆ =
(4) none of these
23. Equation of the bisector of obtuse anglebetween the lines
3 4 7 0, 12 5 2 0x y and x y− + = + − = is :
(1) 11 3 9 0x y+ − = (2) 11 3 9 0x y− + =
(3)21 77 101 0x y− + =
(4)21 77 101 0x y+ − =24. The line L has intercepts a,b on the coordinate
axes . When keeping the origin fixed , thecoordinate axes are rotated through a fixedsame angle , then the same line has intercepts pand q on the rotated axes .Then :
37. If two vertices of an equillateral triangle haveintegral coordinate , then the third vertex willhave :(1) integral coordinates
(2) coordinates which are rational
(3) atleast one coordinate irrational
(4) nothing can be said
38. Equation of the straight line making equalangles with the straight lines
1 0 2 3 1x y and x y+ + = + = and passing
through the point (1,2) may be given as :
(1) ( )13( 3) 2 2 3 8x y x y+ − = ± + −
(2) ( )13( 1) 2 2 3 6x y x y+ + = ± + +
(3) ( )13( 5) 2 2 3 9x y x y+ + = ± + +
(4) none of these
39. A variable line drawn through the point (1,3)meets the x- axis at A and y - axis at B.If therectangle OAPB is completed , where ‘O ’ isthe origin , then locus of ‘P’ is :
(1)1 3
1y x
+ = (2)1 3
1x y
+ =
(3) 3 1x y+ = (4) 3 1x y+ =40. The point (4,1) undergoes the following three
transformations successively :(i) reflection about the liney x=(ii) transformation through the a distance 2 units
along the positive direction of x -axis
(iii) rotation through an angle of 4
π about the
origin in the anti clock wise direction
The final position of the points given by thecoordinates
(1)1 7
,2 2
(2) ( )2,7 2−
(3)1 7
,2 2
− (4) ( )2,7 2
41. The equation of the lines through the points
( )2,3 and making an intercept of length 2 units
between the lines 2 3y x+ = and 2 5y x+ = ,
are :
(1) 3 0,3 4 12x x y+ = + =
(2) 2 0,4 3 6y x y− = − =
(3) 2 0,3 4 18x x y− = + =(4) none of these
42. ABC is a right angled isocelles triangle , rightangled at A (2,1) .If the equation of side BC
is2 3,x y+ = then the combined equation of
lines AB and AC is :
(1) 2 23 3 8 20 10 25 0x y xy x y− + + + + =
(2) 2 23 3 8 20 10 25 0x y xy x y− + − − + =
(3) 2 23 3 8 10 15 20 0x y xy x y− + + + + =(4) none of the above
43. A rectangle ABCD has its side AB parallel toline y x= and vertices A,B and D lie on
1, 2 2y x and x= = = − respectively . Locus
of vertex ‘C ’is :
(1) 5x = (2) 5x y− =
(3) 5y = (4) 5x y+ =44. If the sum of the distances of a point from two
perpendicular lines in a plane is 1, then its locusis :(1) square (2)circle
(3) straight line (4) two intersecting lines
45. Let P (-1, 0) ,Q (0, 0) and R ( )3,3 3 be three
points . Then ,the equation of the bisector of theangle PQR is :
(1)3
02
x y+ = (2) 3 0x y− =
(3) 3 0x y+ = (4)3
02
x y+ =
46. Area of the parallelogram formed by the lines, 1, , 1y mx y mx y nx y nx= = + = = + equals:
having slope 1/2 that divides the rectangle intotwo equal halves is :
(1) 2 1y x= + (2) 2 1x y= +
(3) 2 1y x+ = + (4) 2 1x y+ = +59. Let A=(1,2),B=(3,4) and let C=(x,y) be a point
such
that ( )( ) ( )( )1 3 2 4 0.x x y y− − + − − = If
ar( ABC∆ ) =1, Then max.no.of positions of Cin the xy-plane is :(1) 2 (2) 4
(3) 8 (4) none of these
60. If a vertex of an equillateral triangle is originand the side opposite has its equation
1x y+ = ,then the orthocentreof the triangle is:
(1)1 1
,3 3
(2)2 2
,3 3
(3)2 2
,3 3
(4) none of these
61. ABC is an equillateral triangle such that thevertices B and C lie on two parallel lines at adistance 6 .IF A lies between the parallel linesand at a distance 4 units from one of them thenthe length of the side of the equillateral triangleis :
(1) 8 (2)88
3
(3)7
43
(4) none of these
62. If a,b,c are any three terms of an A.P. ,then
the line 0ax by c+ + = :
(1) has a fixed direction
(2) always passes through a fixed point
(3) always cuts intercepts on the axes such thattheir sum is zero
(4) Forms a triangle with the axes whose area isconstant
63. A family of lines is given by
( ) ( )1 2 1 0,x yλ λ λ λ+ + − + = being the
parametre .The line belonging to this family atthe maximum distance from the point (1,4) is :
(1) 4 1 0x y− + = (2) 33 12 7 0x y+ + =
(3) 12 33 7x y+ = (4) none of these
64. The range of values of the ordinate of a pointmoving on the line x=1,and always remaining inthe interior of the triangle formed by the lines
,y x= x-axis and 4x y+ = is :
(1) (0,1) (2) [0,1]
(3) [0,4] (4) none of these
65. A variable line through the points (a,b) cuts theaxis of reference at A,B respectively .The linesthrough Aand B parallel to the y-axis and x- axisrespectively meet at Phas the equation :
(1) 1x y
a b+ = (2) 1
x y
b a+ =
(3) 1a b
x y+ = (4) 1
b a
x y+ =
66. If the line 1x y
a b+ = moves such that
2 2 2
1 1 1a b c
+ = where c is a constant ,then the
locus of the foot of the perpendicular from theorigin on the line is :(1) a parabola (2) a circle
(3) an ellipse (4) none of these
67. If a line segment AM =a moves in the xy planeremaining parallel to OX ,so that left end point
A slides along the circle 2 2 2,x y a+ = then the
locus of M is :
(1) 2 2 22 ,x y a+ = (2) 2 2 2x y ax+ =
(3) 2 2 2x y ay+ =
(4) 2 2 2 2 0x y ax ay+ − − =68. Equations
( ) ( ) ( )( ) ( )3 3 3 3 3 3
0
( ) 0
b c x c a y a b and
b c x c a y a b
− + − + − =
− + − + − =
(a,b,c are not all equal) will not represent sameline if :
(1) b c= (2) a c=(3) 0a c+ = (4) 0a b c+ + =
69. A=(-4,0), B=(4,0),M,N are variable points on y
maximum of d(P,BC)is : (d(P,BC) representsthe distance between P and BC)
(1) 1 (2)1
2
(3) 2 (4) none of these
110. A line 1L =0 passing through P(1,2) has
equation of its two bisectors with respect to
other line ( )2 0L = as 1 3 4 7 0B x y≡ − − =
and 2 4 3 2 0B x y≡ + − = .Then which of the
following is true ?
(1) 1B is the acute angle bisector
(2) 2B is acute angle bisector
(3) 1 2both B and B (4) nothing can be said
111. A straight line 1L makes an angle 1 4
tan3
− with
parallel lines
3 4 24 0 3 4 12 0x y and x y− − = − − = .If
1L makes an intercept of 3 units with these
parallel lines , then the equation of 1L may be
given as :
(1) 1x = (2) 1y =
(3) 2 3x y= + (4) none of these
112. A ray of light is sent along the line 1x y+ = ,
after being reflected from the line 1y x− = it
is again reflected from the line 0y = ,then the
equation of the line representing the ray aftersecond reflection may be given as :
(1) 1x y+ = (2) 1x y− =
(3) 1y x− = (4) none of these
113. consider the family of lines ( )1 1y y m x x− = −
,in which m is constant and 2 21 1 1x y+ = ,then
all the lines of the family are :(1) concurrent at origin
(2) normal lines to the curve 2 21 1 2x y+ =
(3) tangent lines to the circle 2 21 1 1x y+ =
(4) none of these
114. If the distanceany point (x,y) from the origin isdefined as d(x,y)=max.{|x|,|y|},then the locus ofthe point (x,y),where d(x,y)=1 is :(1) a circle (2) a square
(3) a triangle (4) none of these
115. For the points P=(( ) ( )1 1 2 2, ,P x y and Q x y=of the coordinates plane , a new distance d(P,Q)is defined by
( ) 1 2 1 2, | | | | .d P Q x x y y= − + − Let
O=(0,0) and A=(3,2) .Then the set of points inthe first quadrant which are equidistant (withrespect to the new distance) from 0 and A :(1) consists only one point
(2) consists of the union of a line segment of fi-nite length and an infinite ray
(3) consists of the union of line segments of aconvex polygon
(4) none of the above
116. Let ( ) ( ), ,A a b and B c d= = where
0, 0.c a and d b> > > > Then point C on
the x-axis such that AC+BC is the minimum, is:
(1)bc ad
b d
−− (2)
ac bd
b d
++
(3)ac bd
b d
−− (4)
ad bc
b d
++
117. Let ( ), min.d P OA ≤
( ) ( ) ( ){ }, , , , ,d P AB d P BC d P OC where d
denotes the distence from the point to thecorresponding line and S be the regionconsisting of all those points P inside therectangle OABC such that
( ) ( ) ( )0,0 , 3,0 , 3,2O A B= = = a n d
( )0,2 ,C = which satisfy the above relation,
then area of the region S is
(1) 2 .sq units (2) 3 .sq units
(3) 4 .sq units (4) non of these
118. Two of the lines represented by the equation4 3 2 2 3 4 0ax bx y cx y dxy ay+ + + + =
will bisent the angles between the other two, if :
3. The number of integral points (integral pointmeans both coordinates should be integers)exatly in the interior of the triangle with the
vertices ( ) ( ) ( )0,0 , 0,21 21,0and is :
(1) 133 (2) 190
(3) 233 (4) 105
4. Orthocentre of triangle with the vertices
( ) ( ) ( )0,0 , 3,4 4,0and is :
(1)5
3,4
(2) ( )3,12
(3)3
3,4
(4) ( )3,9
5. The area of the parallelogram formed by thelines
, 1, 1y mx y mx y nx and y nx= = + = = +equals :
(1) ( )2
| |m n
m n
+− (2)
2
| |m n+
(3)1
| |m n+ (4)1
| |m n−
6. The number of integral values of m for whichthe x- coordinate of intersection of the lines
3 4 9 1x y and y mx+ = = + also an integer is
:
(1) 2 (2) 0
(3) 4 (4) 1
7. Let PS be the median of the triangle withvertices P (2,3) ,Q (6,–1) and R(7,3) .Theequation of the line passing through (1,–1) andparallel to PS is :
(1) 2 9 7 0x y− − = (2) 2 9 11 0x y− − =
(3) 2 9 11 0x y+ − = (4) 2 9 7 0x y+ + =
8. If ( )2,a a falls inside the angle made by the
lines , 0 3 , 02
xy x and y x x= > = > ,then a
belongs to :
(1) ( )3,∞ (2)1
,32
(3)1
3,2
− − (4)
10,
2
9. A straight line through the point A(3,4), is suchthat its intercepts between the axes is bisectedat A . Its equation is :
(1) 3 4 7x y− + (2) 4 3 24x y+ =
(3) 3 4 25x y+ = (4) 7x y+ =
10. The line parallel to the x-axis passing throughthe ax+2by+3b=0 and bx-2ay-3a=0 where(a,b) ≠ (0,0) is:
47. Orthocentre of the triangle formed by the linesx + y = 1 and xy = 0 is :
(1) (0,0) (2) (0,1)
(3) (1,0) (4) (–1,1)
48. A ray of light passing through at point (1,2) isreflected on thwe x-axis is point P and passesthrough the point (5,8) .Then the absissa of thepoint P is :
1 0xy x y− − + = and the line 2 3 0ax y+ − =are concurrent, then a is equa to :
(1) –1 (2) 0
(3) 3 (4) 1
71. Distance betweeen the parallel lines
3 4 7 0x y+ + = and 3 4 5 0x y+ − = is :
(1) 2/5 (2) 12/5
(3) 5/12 (4) 3/5
72. Angle between the lines 2 15 0x y− − = and
3 4 0x y+ + = is :
(1) 90° (2) 45°
(3) 180° (4) 60°
73. Equation of a line passing through ( )1, 2− and
perpendicular to the line 3 5 7 0x x− + = is :
(1) 5 3 1 0x y+ + = (2) 3 5 1 0x y+ + =
(3) 5 3 1 0x y− − = (4) 3 5 1 0x y− + =
74. The number of the straight lines which isequally inclined to both the axes is :
(1) 4 (2) 2
(3) 3 (4) 1
75. The image of the point ( )4, 3− with respect to
the line y = x is :
(1) ( )4, 3− − (2) ( )3,4
(3) ( )4,3− (4) ( )3,4−
76. The triangle formed by the lines
0, 3 4, 3 4x y x y x y+ = + = + = is :
(1) isosceles (2)equilateral
(3) right-angled (4)none of these
77. The equation of line perpendicular to x = c is :
(1) y = d (2) x = d
(3) x = 0 (4) none of these
78. Equation of a line passing through the line ofintersection of lines
0, 3 4, 3 4x y x y x y+ = + = + = and
perpendicular to 6 7 3 0x y− + = , then its
equation is :
(1) 119 102 125 0x y+ + =
(2) 119 102 125x y+ =
(3) 119 102 125x y− =
(4) none of these
79. Two lines are drawn through (3, 4), each ofwhich makes angle of 45° with the line x - y =2,then area of the triangle formed by these lines is:
(1) 9 sq. units (2) 9/2 sq. units
(3) 2 sq. units (4) 2/9 sq. units
80. Two consecutive sides of a parallelogram are4x+5y=0 and 7x +2y = 0. One diagonal of theparallelogram is 11x + 7y = 9. If the otherdiagonal is ax + by + c = 0, then :
(1) a = –1, b = –1, c = 2
(2) a =1, b = –1, c = 0
(3) a = –1, b = –1, c = 1
(4) a =1, b =1, c = 1
(5) a = –1, b = –1, c = 1
81. If the lines 3x + 4y + 1 =0, 5 3 0x yλ+ + =and 2x + y -1 = 0 are concurrent, then λ isequal to :
(1) –8 (2) 8
(3) 4 (4) –4
(5) none of these
82. If ( )4,5− is one vertex and 7 8 0x y− + = is
one diagonal of a square, then the equation ofsecond diagonal is :
(1) x + 3y = 21 (2) 2x -3y =7
(3) x +7y =32 (4) 2x +3y =21
(5) x - 3y =21
83. The angle between the pair of straight lines
( )2 2 2 2 2sin sin cos 1 0y xy xθ θ θ− + − = is :
125. Equation to the straight line cutting off anintercept 2 from the negative direction of theaxis of y and inclined at 30° to the positivedirection of axis of x, is
(1) 3 0y x+ − = (2) 2 0y x− + =
(3) 3 2 0y x− − = (4) 3 2 3 0y x− + =
126. If the equation
2 212 10 2 11 5 0x xy y x y k− + + − + =represents two straight lines, then the value of kis
(1) 1 (2) 2
(3) 0 (4) 3
127. The equation of the straight line which isperpendicular to y = x and passes through (3, 2)is
(1) 5x y− = (2) 5x y+ =
(3) 1x y+ = (4) 1x y− =
128. The equation of the straight line joining theorigin to the point of interection of
7 0y x− + = and 2 2 0y x+ − = is
(1) 3 4 0x y+ = (2) 3 4 0x y− =
(3) 4 3 0x y− = (4) 4 3 0x y+ =
129. If we reduce 3 3 7 0x y+ + = to the form
cos sin ,x y pα α+ = then the value of p is
(1)7
2 3(2)
7
3
(3)3 7
2(4)
7
3 2
130. The distance of the point (–2, 3) from the line
5x y− = is
(1) 5 2 (2) 2 5
(3) 3 5 (4) 5 3
131. The equation of straight line through the
intersection of the lines 2 1x y− = and
3 2x y+ = and parallel to 3 4 0x y+ = is
(1) 3 4 5 0x y+ + = (2) 3 4 10 0x y+ − =
(3) 3 4 5 0x y+ − = (4) 3 4 6 0x y+ + =
132. The equation 2 2 5 7 6 0x kxy y x x+ + − − + =represents a pair of straight ines, then k is
(1)5
3(2)
10
3
(3)3
2(4)
3
10
133. The gradient of one of the lines of2 22 0ax hxy by+ + = is twice that of the other,