19.

The medians of a triangle meet at (0 , - 3) and two vertices

are at (- 1 , 4) and (5 , 2). Then the third vertex is at
1) (4 , 15) 2) (- 4 , - 15) 3) (-4 , 15) 4) (4 , - 15)
20. The vertices of a triangle are (0 , 0), (3 , 0) and (0 , 4). The
Multiple choice Questions centroid of the triangle is
LEVEL-I
1   4
1. The distance of the point (x , y) from x-axis is 1)  , 2  2) 1,  3) (0 , 0) 4) (1 , 1)
2   3
1) x 2) y 3) x 4) y
21. The mid-points of AB, BC, CA of a ABC are (6 , - 1), (- 4 , -
2. The distance of the point (x , y) from y-axis is
3), (2 , - 5) respectively. The centroid of ABC is
1) x 2) y 3) x 4) y
3. The locus of a point such that the ratio of its distance from 4  4   4 
1) (4 , 1) 2)  ,  3  3)  , 3  4)   , 3 
two given points is a constant is  3   3   3 
1) a parabola 2) an ellipse 3) a circle 4) a hyperbola 22. If the centroid of a triangle is (3 , 1) and one of the vertices
4. The Distance between the points (2, cot  ) and (1 , 0) is (0 , - 3), then the length of the median is
1) cos ec 2) sec 3) sec 4) cos ec 15 10
1) 5 2) 3) 15 4)
5. The triangle whose vertices are (0 , 0) (2 , 0) and (0 , 3) is 2 3
1) equilateral 2) right angled 23. In the triangle ABC, the co-ordinates of B, C and the cen-
3) isosceles 4) acute angled. troid G are (2 , 3), (- 3 , 4) and (- 5 , - 1) respectively.
6. The points (1 , 5), (- 7 , 9) and (- 10 , - 17) form Then the co-ordinates of A are
1) an acute angled triangle 2) a right angled triangle 1) (14 , 10) 2) (14 , - 10)
3) an obtuse angled triangle 4) a straight line 3) (- 14 , 10) 4) (- 14 , - 10)
7.  
The points (1 , 1), (- 1 , - 1) and  3 , 3 are the vertices of 24. The points (-2 , -5) , (2 , -2), (8 , a) are collinear, then the
1) an equilateral triangle 2) a scalene triangle value of a is
3) an isosceles triangle 4) a right angled isosceles triangle 5 5 3 1
1) 2) 3) 4)
8. The point (22 , 23) divides the join of P (7 , 5) and Q exter- 2 2 2 2
nally in the ratio 3 : 5, then Q = 25. The area of the triangle with vertics at (- 4 , 1), (1 , 2), (4 , 3)
1) (3 , 7) 2) (- 3 , 7) is
3) (3 , - 7) 4) (- 3 , - 7) 1) 17 2) 16 3) 15 4) 1
9. The join of (- 3 , 2) and (4 , 6) is cut by x-axis in the ratio 26. If the points (7 , k), (- 2 , 3) and (- 1 , - 5) are collinear then k
1) 2 : 3 internally 2) 1 : 2 externally is
1
3) 1 : 3 externally 4) 3 : 2 internally. 1) 15 2) 3) 69 4) - 69
7
10. The x-axis divides the line joining the points 27. The area of the triangle formed by the points (a , b + c), (b ,
(5 , 7) and (- 1 , 3) in the ratio c + a) and (c , a + b) is
1) 7 : 3 2) 7 : - 3 3) 6 : 5 4) 6 : - 5 1) abc 2) 0 3) a 2b 2 c 2 4) 1
11. y-axis divides the line joining the points (3 , 6) and (12 , - 3) 28. If the points t1  t2  at12 , 2at1  ,  at22 , 2 at2  and (a , 0) are col-
in the ratio linear, then t1t2 is
1) 1 : 4 2) 1 : - 4 3) 2 : 5 4) 2 : - 5 1) 0 2) 1 3) - 1 4) 2
12. The ratio in which the point (- 1 , 4) divides the line joining 29. The points (- a , - b) (0 , 0) (a , b) and  a 2 , ab  are
(- 7 , 1) and (3 , 6) is 1) collinear 2) vertices of a parallelogram
1) 2 : 1 2) 3 : 1 3) 1 : 2 4) 3 : 2 3) vertices of a rectangle 4) none
13. The ratio in which (- 3 , 4) divides the line joining (1 , 2) and 30. The points (t , 2t), (- 2 , 6) and (3 , 1) are collinear then t =
(7 , - 1) is 3 4
1) 2 : 5 externally 2) 5 : 2 internally 1) 2) 3) 3 4) 4
4 3
3) 1 : 5 externally 4) 1 : 5 internally 31. If the area of the triangle with vertices (x , 0), (1 , 1) and (0
14. The fourh vertex of the rectangle whose other vertices are , 2) is 4 sq.units, then a value of x is
(4 , 1), (7 , 4), (13 , - 2) is 1) - 2 2) - 4 3) - 6 4) 8
1) (10 , 5) 2) (10 , - 5) 3) (8 , 3) 4) (8 , - 3) 32. The point moves such that the area of the triangle form-ed
15. The fourth vertex of the square formed by points by it with the points (1 , 5) and (3 , - 7) is 21 sq.units. The
(2 , 1), (4 , 3), (- 2 , 5) is locus of the point is
1) (2 , 3) 2) (- 3 , 3) 3) (- 4 , 3) 4) (4 , 3). 1) 6x + y - 32 = 0 2) 6x - y + 32 = 0
16. If D (1 , 2), E (4 , 3), F (6 , 4) are the mid - points of the sides 3) x + 6y - 32 = 0 4) 6x - y - 32 = 0
BC , CA , AB respectively of ABC , then A = , B = , C = 33 The locus of the point which is equidistant from (1 , 2) and
1) (6 , 5), (2 , - 2), (1 , 1) 2) (6 , 6), (2 , 2), (1 , 1) (4 , 7) is
3) (9 , 5), (3 , - 3), (- 1 , 1) 4) (9 , 5), (3 , 3), (- 1 , 1). 1) 3x + 5y - 30 = 0 2) 3x - 5y = 30
17. Three vertices of a rectangle are (2, 1),(- 4 , - 1) and (-2 , -3) 3) 3x + 5y + 30 = 0 4) 3x - 5y + 30 = 0
then the co-ordinates of the fourth vertex are 34. The locus of the point which moves so that the sum of its
1) (4, 1) 2) (-4 , - 1) 3) (- 4 , 1) 4) (4 , - 1) distances from the points (a , 0) and (- a , 0) is 2c is
18. Mid points of the sides AB and AC of a triangle ABC are (- x2 y2
x2 y 2
3 , - 5) and (3 , 3) respectively then the length of BC is 1)  1 2) c 2  c 2  a 2 1
c 2 c2
1) 15 2) 10 3) 20 4) 30
 x y 1) (10 , 4), (- 4 , 6), (2 , - 2) 2) (10 , 4), (4 , - 6), (2 , 2)
3) x  y  c 4) c  c  a  1
3) (10 , -4), (- 4 , 6), (2 , 2) 4) (10 , 4), (4 , - 6), (2 , - 2)
35. The locus of the point which moves on the line joining the 15. The extremities of a diagonal of a parallelogram are (3 , -
points (2 , - 3) and (3 , 2) is 4) and (- 6 , 5). If the third vertex is (- 2 , 1) then the fourth
1) - 5x + y = 0 2) - 5x + y - 13 = 0 vertex is
3) 5x + y + 13 = 0 4) 5x - y - 13 = 0 1) (1 , 0) 2) (- 1 , 0) 3) (1 , 1) 4) (- 1 , - 1)
LEVEL-II 16. The centroid of the triangle formed by the points
1. The locus of a point which moves so that its distance from (1 , a), (2 , b) and ( c 2 , - 3) lies on the x-axis, then
the y-axis is three times its distance from the x-axis is 1) a = 3 2) b = 3
1) x = 3y 2) y = 3x 3) y 2  9 x 4) x 2  9 y 2  0 3) a + b = 3 4) a - b = 3
2. The locus of the point which moves so that the square of 17. The centroid of the triangle ABC is (2 , 3) and
its distance from the point (3 , 0) is equal to 7 is A = (4 , 2), B = (4 , 5), then the area of the triangle ABC
1) x 2  y 2  2 2) x 2  y 2  6 x  0 1) 6 2) 9 3) 8 4) 5
3) x  y  6 x  2  0
2 2
4) x 2  y 2  4 18. If the co-ordinates of points A , B , C and D are (6 , 3), (-3 ,
3. If a , b , c are in A.P a , x , b are in G.P and b , y , c are in G.P DBC 1
5), (4 , - 2) and (x , 3x) respectively and if  , then x
the point (x , y) lies on ABC 2
1) a straight line 2) a circle is
3) an ellipse 4) a hyperbola 8 11 7
1) 2) 3) 4) 0
4. The locus of the point equidistant from (1 , - 1) and (- 1 , 1) 11 8 9
is 19. K  0 If the three points (k , 2k), (3k , 3k) and (3 , 1) are
1) x + y = 0 2) x - y = 0 collinear then k is
3) 2y - x = 0 4) x + 2y = 0 1 1
1) 3 2) - 3 3) 4) 
5. A line is of the length 10 units and one end is at (2 , -3). If 3 3
the abcissae of the other end is 10, then its ordinate is 20. The end points of the base of a triangle are (- 1 , 2) and (3 ,
1) 2 , 7 2) 7 , 2 3) -3 , 9 4) -9 , 3 5). The locus of the vertex, if the altitude is 7 is
6. Three points (0 , 0), (3 , 3 ), (3,  ) form an equilateral tri- 1) 3x - 4y + 46 = 0 2) 3x - 4y + 24 = 0
angle. Then  = 3) 3x + 4y + 46 = 0 4) 3x - 4y - 46 = 0
1) 2 2) - 3 3) - 4 4)  3 21. A point P moves in a plane such that the algebraic sum of
7. If (9 , - 9), (1 , - 3) are the ends of a right angle isosceles its distances from perpendicular lines is equal to a, then
triangle, then the third vertex is the equation of its locus is
1) (8 , - 2) 2) (- 8 , 2) 3) (8 , - 8) 4) (8 , 8) 1) x + y = a 2) x + y + a = 0
8. If A (a , a), B (- a , - a) are two vertices of an equilateral 3) x + y = 2a 4) x  y  a 2
triangle, then its third vertex is 22. If A = (6 , 0) and B = (0 , 4) and O is the origin then the locus
a 3 a 3 of P such that POB  2POA
1)  2 ,  2 
 

2)  a 3 , a 3  POA is
1) x 2  3 y 2  0 2) x 2  3 y 2  0

3) a 3 , a 3  4)   a 3, a 3  3) x  9 y  0
2 2
4) none
9. If the point (1 , 1) is equidistant from the point (a + b , b - a)  1 2 1 1
and (a - b , a + b) then 23. Is A   t 2 , 2t  and B   2 ,   and S  1, 0  , then  
 t t  SA SB
1) a + b = 0 2) a + b = 1 1 1
1) 2 2) 3) 4) 1
3) a = b 4) b - a = 1 2 5
24. Area of the triangle with vertices (a , b) , ( x1 , y1 ) and
 8
10. The points A  0,  B 1,3 and C  82,30  are the vertices of ( x2 , y2 ) where a , x1 , x2 are in G.P, with common ratio r and
 3
1) an acute angled triangle 2) an isoscles triangle b , y1 , y2 are in G.P. With common ratio s is
3) a right angled triangle 4) none 1
1) ab (r - 1) (s - 1) (s - r) 2) ab (r  1) ( s  1) ( s  r )
11. The base vertices of a right angled isosceles triangle are (7 2
1
, 9) and (3 , - 7), then the third vertex is 3) ab ( r  1) ( s  1) ( s  r ) 4) ab (r + 1) (s + 1) (r - s).
2
1) (13 , 1) or (- 3 , 3) 2) (13 , - 1) or (3 , - 3) 25. Locus of centroid of the triangle whose vertices are (a cos t
3) (13 , - 1) or (- 3 , 3) 4) (13 , 1) or (3 , 3) , a sin t),(b sin t , - b cos t) and (1 , 0), where t is a parameter
12. The harmonic conjugate of (1 , 4) with respect to the point is
(- 1 , 2) and (4 , 7) is 1)  3x  1   3 y   a 2  b 2
2 2
2)  3x  1   3 y   a 2  b 2
2 2

1) (11 , 8) 2) (- 11 , 8) 3)  3x  1   3 y   a  b
2 2 2 2
4)  3x  1   3 y   a 2  b 2
2 2

3) (- 11 , - 8) 4) (11 , - 8) 26. If the equation of the locus of a point equidistant from the
13. If P and Q are the points (-3 , 4) and (2 , 1) then the coordi- points  a1 , b1  and  a2 , b2  is  a1  a2  x   b1  b2  y  c  0 , then
nates of the point R on PQ produced such that PR = 2QR the value of ‘c’ is
are 1 2 2 2 2
1) a12  b12  a22  b22 2)  a2  b2  a1  b1 
 1 5 2
1) (2 , 4) 2) (3 , 7) 3) (7 , - 2) 4)   ,  1 2
 2 2 4)  a1  a2  b1  b2 
2 2 2
3) a12  a22  b12  b22
14. If the midpoints of the sides of a triangle are 2
27. The locus of the point x  a cos  , y  b sin  is
(- 1 , 2), (6 , 1) and (3 , 5) then the co-ordinates of the verti-
ces of the triangle are
 x 2 y2 x 2 y2 
1)  1 2)  1 3x-5y+7=0 is then one of the value of k=
a 2 b2 a 2 b2 4
x y2
2
1)1 2)2 3)3 4)4
3) 2  2  2 4) a 2 x 2  b 2 y 2 1
a b 7. Perpendicular distance from the origin to the line joining
28. The locus of the point x  a  b cos  , y  b  a sin  is the points (acos  ,asin  ) (acos  ,asin  ) is
1) circle 2) ellipse 3) parabola 4) hyperbola   
29. The locus of the point  a cos3  , a sin 3   is 1) 2a cos (  -  ) 2) a cos 
 2 

2 2 2 2 2 2
1) x 3  y 3  a 3 2) x 3  y 3  a 3      
2
3) x 3  y 3  a 2
2 3
4) x 2  y 2  a 2
3 3 3 3) 4a cos   4) a cos  
 2   2 
30. A line segment AB of length ‘a’ moves with its ends on the 8. The equation of the base of an equilateral triangle is
axes. The locus of the point P which divides the line in the x+y-2=0 and the vertex is (2,-1) then the length of side is
ratio 1 : 2 is
2
1) 9x 2  4y 2  a 2 2) 9  y 2  4x 2   4a 2 1)1 2)2 3)3 4)
3
3) 9  x  4y   4a
2 2 2
4) 9x 2  9y 2  4a 2 9. A line passing through P(-2,3) meets the axes in A and B.
31. The distance between the points (a cos 48 , 0) and If P divides AB in the ratio of 3:4 then the perpendicular
( 0 , a cos12 ) is ‘ d ‘ then d 2  a 2  distance from (1,1) to the line is

1)
a2  5 1  2)
a2  5 1 3) a  2
5 1  4) a 2
5 1  1)
9
2)
7
3)
8
4)
6
4 4 8 8 5 5 5 5
32. The centroid of a triangle formed by the points (0 , 0) 10. One side of an equiateral triangle is 3x+4y=7 and its
(cos  ,sin  ) and (sin  ,  cos  ) lie on the line y = 2x, then  vertex is (1,2). Then the length of the side of the triangle
is is
1 1 1 1 4 3 3 3 8 3 4 3
1) Tan 1 2 2) Tan 3) Tan 4) Tan 1 ( 3) 1) 2) 3) 4)
3 2 17 16 15 15
33. The maximum area of the triangle formed by the points 11. If p, q are the perpendicular distances from the origin to
(0 , 0) , (a cos  , b sin  ) and (a cos  ,  b sin  ) the lines xsec  + ycosec  = a and xcos  – ysin  =
(in square units) acos2  then 4p2+q2 =
3ab ab a2
1) 2) ab 3) 4) a 2b 2 1) a2 2) 4a2 3) 2a2 4)
4 2 2
34. If the lines 3x+2y-5=0, 2x-5y+3=0, 5x+by+c=0 are concur- 12. A : The distance between the St.lines y=mx+c1, y=mx+c2
rent then b+c = is |c1-c2| then m =
1) 7 2) -5 3) 6 4) 9 1)0 2)1 3)2 4)3
35. The straight lines x+2y-9=0, 3x+5y-5=0 and ax+by-1 are
13. The distance between two parallel lines is
concurrent if the straight line 22x-35y-1=0 passes through
the point p1-p. Equation of one line is xcos  + ysin  = p then the
1) (a, b) 2) (b,a) 3) (-a,b) 4)(-a, -b) equation of the 2nd line is
1) xcos  + ysin  + p1 + 2p = 0
LEVEL-III 2) xcos  + ysin  = 2p1 - p
1. If a  b  c and if ax  by  c  0 bx  cy  a  0 and 3) xcos  + ysin  = 0
cx  ay  b  0 are 4) xcos  + ysin  + p1 - 2p = 0
concurrent. Then the value of 2a2b1c1 2b2c1a12c2a1b1 14. The perpendicular distance from the point of intersection
1) 1 2) 4 3) 8 4) 16 of the lines 3x+2y+4=0,
2. The acute angle between the lines 2x+5y-1=0 to the line 7x+24y-15=0
lx + my = l+m, l (x-y) + m (x+y) = 2m is 2 1 1 2
1) 2) 3) 4)
    3 7 5 5
1) 2) 3) 4)
15. The foot of the perpendicular drawn from (0, 0) to the
4 6 2 3
3. The lines (a+b)x+(a-b) y – 2ab = 0, line xcos  + ysin  – p = 0 is
(a-b)x + (a+b) y – 2ab = 0 and x+y=0 from an isosceles 1) (psin  , pcos  ) 2) (psec  , pcos  )
3) (pcos  , psin  ) 4) (p, p)
triangle whose vertical angle is
16. A line passing through the points (7,2) (-3,2) then the
 1  2 ab 
1) 2) tan  2 2  image of the line in x-axis is
2  a b 
1) y = 4 2) y = 9
1  a  1  a 
3) tan   4) 2 tan   3) y = -1 4) y = -2
b b
4. The angle between the lines kx+y+9=0, y-3x=4 is 45 then O 17. The reflection of the point (6,8) in the line x=y is
the value of k is 1) (4,2) 2) (-6,-8) 3) (-8,-10) 4) (8,6)
1) 2 or ½ 2) 2 or -1/2 18. Image of (1,2) w.r.t. (-2, -1) is
3) -2 of ½ 4) -2 or -1/2 1) (0,5) 2) (-4,-3) 3) (-5,-4) 4)(-4,-5)
5. The angle between the lines x cos  +y sin  = p1 and 19. If 2x+3y+4=0 is the perpendicular bisector of the
x cos  + y sin  = p2 where    is segment joining the points A(1,2) and B (a b) then the
1)    2)    3)  4) 2   value of a+b is
6. If the angle between the lines kx-y+6=0, 81 136 135 134
1)  2)  3)  4) 
13 13 13 5
20. P is the midpoint of the part of the line 3x+y-2=0 34. An equilateral triangle has its centroid at origin and one
intercepted between the axes. Then the side is x+y-1=0 then the vertex of the triangle not on the
image of P in origin is given side is
 1  1   1  1) (1, 1) 2) (-1, -1) 3) (2, 2) 4)(-2, -2)
1)  1,   2)   , 4  3)   , 1 4) (-2, -3)
 3  3   3  35. Area of the quadrilateral formed by the lines 2x-5y+7=0,
21. L1 and L2 are two intersecting lines. If the image of L1 w.r.t. 5x+2y-1=0, 2x-5y+2=0 and 5x+2y+4=0
L2 and that of L2 w.r.t. L1 concide, then the angle between 25 3 2 7
1) 2) 3) 4)
L1 and L2 is 29 4 7 16
1) 35O 2) 60O 3) 90O 4) 45O 36. Two sides of a rectangle are 3x+4y+5-0, 4x-3y+15=0 and
22. The image of the point P (3,5) with respect to the line y = x its one vertex is (0,0). Then the area of the rectangle is
is the point Q and the image of Q with respect to the line 1) 4 2) 3 3) 2 4) 1
y = 0 is the point R (a,b), then (a, b)= 37. If the straight lines y = 4-3x, ay=x+10, 2y+bx+9=0
1) (5,3) 2) (5,-3) 3) (-5,3) 4) (-5,-3) represents three consecutive sides of a rectangle then
the value of ab is
23. If the vertices of an equilateral triangle is the
1) 12 2) 6 3) 18 4) 24
origin and side opposite to it has the equation x+y=1,
38. Area of the quadrilateral formed by the lines 4y-3x-a=0,
then the orthocentre of the triangle is 3y-4x+a=0, 4y-3x-3a=0, 3y-4x+2a=0 is
1 1 2 2 a2 a2 2a 2 2a 2
1)  ,  2)  ,  3)(1,1) 4)(1,3) 1) 2) 3) 4)

3 3  
3 3  5 7 7 9
24 If the circumcentre of the triangle lies at (0,0) and 39 The point (2,3) is reflected four times about co-ordinate
centroid is mid point of the line joining the points (2,3) axes continuously starting with x-axis. The area of
uadrilateral formed in sq.units is
and (4,7),then its orthocentre lies onthe line
1) 24 2) 6 3) 12 4) 5
1)5x-3y=0 2)5x-3y+6=0 3)5x+3y=0 4)5x+3y+6=0 40. The two lines y = 2x, y = -2x are
25. Two sides of a triangle are y = m1x and y=m2x. m1, m2 are 1) Parallel 2) Perpendicular
the roots of the equation x2+ax-1=0. For all values of a, 3) Coincident
the orthocentre of the triangle lies at 4) Equally inclined with the axes
3 3
1) (1, 1) 3)  , 
2) (2, 2) 4) (0,0) IIT CORNER
2 2
26. The equation to the base of an equilateral triangle 1. The total number of circles that can be drawn touching
all the three lines x+y-1=0, x-y-1=0, y+1=0 is
is   
3 1 x  
3  1 y  2 3  0 and opposite vertex is 1) 1 2) 2 3) 3 4) 4
A(1,1) then the Area of the triangle is 2. A ray of light passing through the point (8,3) and is
reflected at (14,0) on x axis. Then the equation of the
1) 3 2 2) 3 3 3) 2 3 4) 4 3
reflected ray
27. The area of the triangle formed by the axes and the
1) x+y=14 2) x-y=14 3) 2y=x-14 4) 3y=x-14
line (cosh  - sinh  ) x+ (cosh  + sinh  ) y=2 in
3. Equation of the line which join the origin and the point
square units is
1) 4 2) 3 3) 2 4) 1 of trisection of the portion of the line x+3y-12=0
28. Area of the triangle formed by the lines x=0, x+y=2, intercepted between the axes is
x-y=1/2 is 1) x=6y 2) x-5y=0 3) 3x-7y=0 4) 2x-5y=0
25 15 3 3 4. The number of circles that touch all the straight lines
1) 2) 3) 4) x+y - 4 = 0, x - y+2 = 0 and y = 2 is
16 8 4 8
29. Area of triangle formed by angle bisectors of coordinate 1) 1 2) 2 3) 3 4) 4
axes and the line x=6 in sq.units is 5. p is the length of the perpendicular drawn from the
1) 36 2) 18 3) 72 4) 9 origin upon a straight line then the locus of mid point
30. A line passing through (3,4) meets the axes OX and OY at of the portion of the line intercepted between the
A and B respectively. The minimum area of the triangle coordinate axes is
OAB in square units is 1 1 1 1 1 2
1)   2)  
1) 8 2) 16 3) 24 4) 32 x2 y2 p2 x2 y2 p2
31. Area of the triangle formed by the lines 7x+3y=35, x=5,
1 1 4 1 1 1
2y-3=0 is 3)   4)  
x2 y2 p2 x2 y 2 p
27 17 13 27
1) 2) 3) 4) 6. The limiting position of the point of intersection of the lines
28 28 25 56
3x+4y=1 and (1+c) x+3c2y=2 as c tends to 1 is:
32. Area of the triangle formed by the lines 7x-2y+10=0,
1) (4, -5) 2) (-5, 4) 3) (5, -4) 4) (-4, 5)
7x+2y-10=0, y+2=0 is
7. The equation of the straight line whose intercepts on x-axis
1) 7 2) 14 3) 18 4) 12
and y-axis are respectively twice and thrice of those by the
33. Area of triangle formed by angle bisectors of coordinate
line 3x + 4y = 12, is
axes and the lineY=6 in sq.units is
1) 9x + 8y = 72 2) 9x - 8y = 72
1)36 2)24 3)72 4)16
3) 8x + 9y = 72 4) 8x+9y+72=0
8. The number of integer values of m, for which x coordinate
of the point of intersection of the lines 3x  4 y  9 and
 y  mx  1 is also an integer, is  1  a  a
1) 2 2) 0 3) 4 4) 1 1)  0,   2)  0,  3)  0,   4)  0, 0 
 2  m  m
9. If thelinesy=mrx, r=1, 2, 3 cut off equal intercepts on the trans- 22. Two equal sides of an isoceles triangle are given by
versal x+y=1, then 1+m1, 1+m2, 1+m3 are in : 7 x  y  3  0 and x  y  3  0 and the third side passes
1) A.P. 2) G.P. 3) H.P. 4)A.G..P through the point ( 1,10 ) the slope m of the third side is
10. y = cos x cos(x+2)-cos2(x+1) is: given by
1) a straight line passing through the origin 1) 3m 2  1  0 2) m 2  1  0
2) a straight line having slope 3 3) 3m 2  8m  3  0 4) m 2  3  0
3) a parabola with vertex (0, -sin21) 23. One vertex of an equilateral triangle is (2,3) and the
4) a straight line parallel to x-axis passing through equation of one side is x-y+5=0 then the equations to the
 2 
other sides are
 ,  sin 1 .
4  
1) y-3 =  2  3 (x-2)  2) y-3 = 2  1 (x-2)  
11. The lne 3x-2y = 24 meets x-axis at A and y-axis at B. The 
3) y-3 = 3  1 (x-2)  4) y-3 = 5  1 (x-2)  
perpendicular bisector of AB meets the line through (0, - 24. In an isosceles triangle, the ends of the base are (2a, 0)
1) and parallel to x-axis at C. Then C is
and (0, a) and one side is parallel to y-axis. The third
 7   15   11   13  vertex is
1)  , 1 2)  , 1 3)  , 1 4)  , 1
 2   2   2   2 
 
5a   5a
12. A straight line through the origin 'O' meets the parallel lines 1)  2a  2)  , a  3) (2a, a) 4)(4a, a)
 2   4 
4 x  2 y  9 and 2 x  y  6  0 at points P and Q respectively.
then point O divides the segment PQ in the ratio 25. The acute angle bisector between the lines 3x-4y-5=0,
1) 1 : 2 2) 3 : 4 3) 2 : 1 4) 4 : 3 5x+12y-26=0 is
13. If the line x  a  m , y  2 and y  mx are concurrent , then 1) 7x-56y+32=0 2) 9x-3y+13=0
least value of a is 3) 14x-112y+65=0 4) 7x-13y+9=0
26. The obtuse angle bisector between the lines 2x-y-4=0, x-
1) 0 2) 2 3) 2 2 4) 2
2y+10=0 is
14. If the lines x+ay+a=0, bx+y+b=0, cx+cy+1=0 (a, b, c being
1) x-y+7=0 2) 3x-y+5=0
distinct and  1) are concurrent then the value of
3) x+y-14=0 4) 2x+3y-5=0
a b c 27. The equation of the bisector of the angle between the
  =
a 1 b 1 c 1
lines x-7y+5=0, 5x+5y-3=0 which is the supplement of
1) -1 2) 0 3) 1 4) 3
the angle containing the origin will be
15. If 4a 2  9b 2  c 2  12ab  0 then the family of straight lines 1) x+3y-2=0 2) x-3y+2=0
ax  by  c  0 is concurrent at 3) 3x-y+1=0 4) 3x+y+2=0
1) (2,3) or (-2,-3) 2) (2,-3) or (-2,6) 28. Equation of the line through the point of intersection of
3)(-2,-4) or (-2,3) 4) (2,5) or (-1,-5) the lines 3x+2y+4=0 and 2x+5y-1=0 whose distance from
(2,-1) is 2.
16. If the point (a, a) falls between the lines |x+y|=2, then:
1) 2x-y+5=0 2) 4x+3y+5=0
1) | a |=2 2) | a |=1
3) x+2=0 4) 3x+y+5=0
1
3) | a |<1 4) | a |< 29. The equation of the line passing through the point of
2
intersection of the lines 2x+y=5 and y=3x-5 and which is
17. In an isosceles triangle OAB, O is the origin and OA = OB
at the minimum distance from the point (1,2) is
= 6. The equation of the side AB is x-y+1=0. Then the area
of the triangle is 1) x+y=3 2) x-y=1 3) x-2y=0 4) 2x+5y=7
30. If the base of an isosceles triangle is of length 2P and
142 71
1) 2 21 2) 142 3) 4) the length of the altitude dropped to the base is q, then
2 2
the distance from the mid point of the base to the side
18. The equation of a straight line L is x+y=2, and L1 is
of the triangle is
another straight line perpendicular to L and passes
pq 2 pq
through the piont (1/2, 0), then area of the triangle 1) p 2  q 2 2) p 2  q 2
formed by the y-axis and the lines L, L1 is
25 25 25 25 3 pq 4 pq
1) 2) 3) 4) 3) 4)
8 16 4 12 p2  q2 p2  q2
19. The abscissa of the orthocentre of the triangle formed
31. If the lines p1x  q1 y  1, p2 x  q2 y  1and p3 x  q3 y  1
1
by the lines y=mix+ (i=1,2,3) is be concurrent, then the points  p1 , q1  ,  p2 , q2  and  p3 , q3  ,
mi
1) 1 2) -1 3) 2 4) -2 1) are collinear
20. The orthocentre of the triangle formed by the lines 2) form an equilateral triangle
x+y=1, 2x+3y=6 and 4x-y+9=0 lies in quadrant number 3) form a scalene triangle
1) 1st 2) IInd 3) IIIrd 4) IVth 4) form a right angled triangle
21. The foot of the perpendicular from (a,0) on the line 32. If the lines x+py+p=0, qx+y+q=0 and rx+ry+1=0 (p,q,r
being distinct and  1) are concurrent, then the value
a
y  mx  is p q r
m
of p  1  q  1  r  1 =
 1)1 2) -1 3) 2 4) -2
KEY
33. If a, b, c are the pth, qth and rth terms of an H.P. then the
LEVEL-I
lines bcx+ py + 1 = 0, cax + qy + 1 = 0 and abx + ry + 1 = 0:
1) are concurrent 2) form a triangle
1 4 2 3 3 3 4 4 5 2
3) are parallel
6 2 7 1 8 4 9 3 10 2
4) mutuvally perpendicular lines 11 2 12 4 13 1 14 2 15 3
34. If x1 , y1 are roots of x 2  8x  20  0, x2 , y2 16 4 17 4 18 3 19 2 20 2
are the roots of 4 x 2  32 x  57  0 and x3 , y3 are the roots of 21 2 22 2 23 4 24 2 25 4
9 x 2  72 x  112  0, then the points  x1 , y1  ,  x2 , y2  and  x3 , y3  26 4 27 2 28 3 29 1 30 2
1) are collinear 31 3 32 1 33 1 34 2 35 4
2) form an equilateral triangle LEVEL-II
3) form a right angled isosceles triangle 1 4 2 3 3 2 4 2 5 4
4) are concyclic 6 4 7 1 8 2 9 3 10 4
35. Let ax + by + c = 0 be a variable straight line, where a,b and 11 3 12 3 13 3 14 1 15 2
c are 1st ,3rd and 7th terms of an increasing A.P. then the 16 3 17 2 18 2 19 4 20 2
variable straight line always passes through a fixed point 21 1 22 3 23 4 24 3 25 3
which lies on. 26 2 27 2 28 2 29 2 30 3
1) x 2  y 2  13 2) x 2  y 2  5 31 4 32 4 33 3 34 2 35 2
3) y  4 x
2
4) 3x  4 y  9 LEVEL-III
36. If t1 and t2 are the roots of the equation t 2   t  1  0 .Where 1 3 2 1 3 2 4 2 5 2
'  ' is an aribitary constant. Then the line joining the point 6 4 7 2 8 4 9 3 10 3
 at1
2
, 2at1  and  at2 , 2at 2  always passes through a fixed
2
11 1 12 1 13 4 14 3 15 3
point 16 4 17 4 18 3 19 1 20 3
1) ( a,0 ) 2) ( -a,0 ) 3) ( 0,a ) 4) ( 0,-a ) 21 2 22 2 23 1 24 1 25 4
37. No.of integral values of 'b' for which the origin and the (1
26 3 27 3 28 1 29 1 30 3
,1) lie on the same side of the straight line a 2 x  aby  1  0 ,
31 4 32 2 33 1 34 2 35 1
 a  R  {o}is 36 2 37 3 38 3 39 1 40 4
1) 4 2) 3 3) 5 4) 6
IIT CORNER
38. If (a, 2) is a point between the lines 3x  4 y  2 and
1 4 2 3 3 1 4 4 5 3
3 x  4 y  5, then :
1) -2<a<-1 2) a<-1 3) a>-1 4) a=0
6 2 7 1 8 1 9 3 10 4
39. The quadratic equation whose roots are the x and y inter- 11 1 12 2 13 3 14 3 15 1
cepts of the line passing through (1,1) and making a traingle 16 3 17 4 18 2 19 2 20 2
of area A with the co -ordinate axes is 21 2 22 3 23 1 24 1 25 3
1) x 2  Ax  2 A  0 2) x 2  2 Ax  2 A  0 26 3 27 1 28 2 29 1 30 1
3) x 2  Ax  2 A  0 4)( x - A)(x+A)=0
31 1 32 1 33 1 34 1 35 1
40. Aera of the parallelogram formed the lines y  mx , 36 2 37 2 38 1 39 2 40 4
y  mx  1 , y  nx , y  nx  1
Hints & Solutions
mn 2
1)  m  n 2 2)  m  n 2 LEVEL-III
1. a+b+c=0
1 1 a 3  b3  c 3  3abc
3)  m  n  4) m  n
m1  m2
2. tan  
1  m1m2

m1  m2
3. tan  
1  m1m2

m1  m2
4. tan   =1
1  m1m2

a1a2  b1b2
5. cos  
a  b12 a22  b22
2
1

m1  m2
6. tan  
1  m1m2

7. Find the distance between  0, 0  and midpoint of

 a cos  , a sin   and  a cos  , a sin  
 36. The perpendicular distance from orign lines are a,b.
3 1
8. a Then area = ab
2 2
37. a  3, b  6
9. The line passing through P is 2 x  y  7  0 38. The area of the parallelogram formed by the lines a1x +
3 4 b1y + c1 = 0, a2x + b2 y + d1=0, a1x + b1y + c2= 0, a2x + b2
10. a
2 5  c1  c2  d1  d 2 
11. Standard result y + d2= 0 is a1b 2  a 2 b1
Sq.units.
c1  c2
12.  c1  c2 39. A  2,3 B  2,3 C  2  3 D  2, 3
1  m2
40. Given lines are equally inclined angular bisectors of
13. Verify the distance between the parllel lines coordinate axes
14. Intersecting point  2,1 IIT CORNER
15. Apply foot of the perpendicular formula 1. Given lines form a triangle
16. Line equation y = 2 2. Write the image of  8,3 in X-axis and write the
Image with respect to x-axes is y = -2
equation through that point and 14,0 
17. The reflection of the point  a, b  in the line x  y is (b,a)
3. Point of trisection is (8,4/3)
18. The image of  x1, y1  w.r.to  x, y  is  2 x  x1, 2 y  y1  equation of the line y = x / 6
19. B is the image of A w. r to 2 x  3 y  4  0 4. Given lines form a triangle
20. The image of (a,b) w.r .to orign is (-a,-b) x y
5. Equate the distance from  0, 0  to the line x  y  2
21.   60 0 1 1

22. Image of P(3,5) w. r to the line y = x is Q(5,3) 3c2  8 5 c

Image of Q(5,3) w. r to the line y =0 is R(5,-3) 6. (B) Solving (1) and (2), x  ,y 2 .
9c2  4c  4 9c  4c  4
1 1 x y
23. Foot of the perpendicular D   ,  7. a  8, b  9 ;  1
2 2   8 9
G(=O) divides median in the ratio 2:1 8. By solving (1) and (2)
24. circumcentre,centroid and orthocentre lies on asame 5
x
line. 3  4m
25. m1m2  1 Given lines are perpendicular  3  4m  1,  1, 5 and -5
26. Area of an equilateral triangle is  4m   2, 4, 2, 8
 m   1,  2 ( m is an integer)
h2
where h is the height of the triangle 9m  3
3 y
3  4m
c2 3  4m  1, 1, 9m  3, 9 m  3
27.   2 ab
1 6
28. Given lines form a right angle triangle m  0, 1, , the satisfaction value of the m = -1,-2
2 13
29. Equations of the angular bisectors of the axes 9. Solving x+ y=1 and y=m1x,
are y  x and y   x
 1 m 
 p, q    3,4  then minimum area = 2pq
1
30. we get: A  1  m , 1  m 
 1 1 
31. The area of the triangle
 1 m2 
a1 b1 c1
2 similarly B is  1  m , 1  m 
 2 2 
1
a2 b2 c2
2  1 m3 
a3 b3 c3 and C is  1  m , 1  m 
 1 2 2 1  a2b3  a3b2  a3b1  a1b3 
a b  a b  3 3 

by the question, B is mid point of AC

2 2 1 1
a1 b1 c1   
1 1  m2 1  m1 1  m3
a2 b2 c2
2  1  m1 ,1  m2 ,1  m3 are in H.P.
32. The area of the triangle a3 b3 c3
 a1b2  a2b1  a2b3  a3b2  a3b1  a1b3  1
10.  [2 cos x cos( x  2)]  cos ( x  1)
2

2
33. Equations of the angular bisectors of the 1 1
 [2 cos 2 ( x  1)  1]  cos 2  cos 2 ( x  1)
axes are y  x and y   x 2 2
34. mid point of the side =foot of perpendicular
= -sin2 1,
1 1 Which is a st. line, parallel to x-axis through
from (0,0) to x+y-1=0is D   ,   2 
 2 2
 , sin 1
G divides median in the ratio 2:1 4 
35. The area of the parallelogram formed by the lines a1x + 11. perpendicular bisector of AB is
b1y + c1 = 0, a2x + b2 y + d1=0, a1x + b1y + c2= 0, a2x + b2 y + 2 x  3 y  10  0............. 1 ;
 c1  c2  d1  d 2  y  1.............  2 
d2= 0 is a1b 2  a 2 b1
Sq.units.
Solve above equations
12. The normal form of 4 x  2 y  9 is a1 x  b1 y  c1 a2 x  b2 y  c2

4x 2y 9 a b
2 2
a22  b22
  and 1 1
20 20 20
2x y 6
28. Point of intersection   2,1  and verification
2 x  y  6  0 is  
5 5 5 29. The point 1, 2  lies on L1   L2  0
P0 9 / 20 9 3
     3:4 30. Consider B   p,0  C  p,0  A  0, q 
Q0 6 / 5 12 4
13. by eliminating x,y from three equations we get p1 q1 1
–2 = m (a + m)  m 2  am  2  0 p2 q2 1  0
31. ,
Since m  R  dis  0 p3 q3 1
 a2  8  0  a  2 2 p1 q1 1
p2  p1 q2  q1 0  0
1 a a
p3  p2 q3  q2 0
b 1 b 0
14. p2  p1 p3  p2
c c 1 
q2  q1 q3  q2
Use det properties
1 p p
15.  2a  3b   c 2  0
2
q 1 q 0
32.
16. From the figure r r 1
1  a  1 i.e. | a | 1.
Y Use det formula
B bc p 1
x+
y=
ca q 1
2
33.
C X ab r 1
O A
x+
y= m  ( p  1) d p 1
-2
m  (q  1)d q 1,
D = abc
m  (r  1)d r 1
17. Let D is the mid point of AB
p p 1
1 71  abcd q q 1
OD  ; AD  =abcd(0)=0.
2 2 r r 1
AB  2 AD 34. x1  10, y1  2
18. Find the area of the triangle formed by the lines
19 3
x y 2 x2  , y1 
2 2
2x  2 y  1
28 4
x0 x3  , y3 
3 3
19. Standard result
35. Let ' d ' be common difference of A.P. then
 7 b  a  2 d , c  a  6d
20. Orthocentre =  1, 
 2   b  a  3  c  a  2a  3b  c  0
 a So the straight line passes through ( 2,-3 ) which also satis-
 am  
21. h  a  k    m fies x 2  y 2  13
m 1 1 m2
36. t1  t2   , t1t2  1 ,

22.
m7

m 1  y  2at1  
2
t1  t2

x  at1
2

1  7m 1 m
 a b 
(or) 3m 2  8m  3  0   y  2 x  2a  1
2 2
ab
 h, k    , 2 2 
a b a b 
2 2

m 1  2 x  a   y  0
23. Let slope of another side is m tan 60 
0
;
1 m
Passes through fixed point ( –a, 0 ).
m  2  3 37. a 2  ab  1  0
24. a  R  D  0  b 2  4  0  2  b  2
A  2a,0  B  0, a  C  2a, k 
 Sine(0, 0) and 1,1  b  1, 0,1
BC  AC
38. (0,0)(a,2) oppiste side of 3x+4y=2
25. a1a2  bb ,
1 2  32  0 c1c2  130  0 39. a  b  ab
a1 x  b1 y  c1 a2 x  b2 y  c2 1
Use  ab  A
a b2
1 1
2
a22  b22 2
40. Area of the parallelogram
26. a1a2  b1b2  4  0
 c1  c2  d1  d 2 
a1 x  b1 y  c1 a2 x  b2 y  c2
Use  a1b2  a2b1
a12  b12 a22  b22

27. a1a2  b1b2  0

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