Straight Lines

If a straight line drawn through the point of intersection of the lines 4x+3y-1=0 and 3x+4y-1=0, meets the co-ordinate axes at the points P and Q, then the locus of the mid point of PQ is :

1. x+y-7=0
2. x+y-14xy=0
3. 2x+y+14xy=0
4. x+2y-14xy=0

In an equilateral triangle PQR, let the vertex P be at (3, 5) and the side QR be along the line x+y=4. If the orthocentre of the triangle PQR is ( , ), then 9( + ) is equal to:

1. 16
2. 27
3. 36
4. 48

Let the line L1 : x+3=0 intersect the lines L2 : x-y=0 and L3 : 3x+y=0 at the points A and B, respectively. Let the bisector of the obtuse angle between the lines L2 and L3 intersect the line L1 at the point C. Then BC2 : AC2 is equal to:

1. 5:1
2. 1:5
3. 2:3
4. 3:2

Let the vertex A of a triangle ABC be (1, 2), and the mid-point of the side AB be (5, -1). If the centroid of this triangle is (3, 4) and its circumcenter is ( , ), then 21( + ) is equal to :

1. 309
2. 403
3. 497
4. 524

Let the mid points of the sides of a triangle ABC be ( 52, 7 ), ( 52, 3 ) and (4, 5). If its incentre is (h, k), then 3h+k is equal to :

1. 11
2. 12
3. 13
4. 14

Let ABC be an equilateral triangle with orthocenter at the origin and the side BC on the line x + 2 2y = 4. If the co-ordinates of the vertex A are (α, β), then the greatest integer less than or equal to | + 2 | is

1. 5
2. 4
3. 2
4. 3

Let A(1, 0), B(2, -1) and C ( 73, 43 ) be three points. If the equation of the bisector of the angle ABC is x + y = 5, then the value of 2 + 2 is

1. 5
2. 10
3. 8
4. 13

Let the angles made with the positive x-axis by two straight lines drawn from the point P(2, 3) and meeting the line x + y = 6 at a distance 23 from the point P be 1 and 2. Then the value of ( 1 + 2) is:

1. 2
2. 3
3. 12
4. 6

A rectangle is formed by the lines x = 0, y = 0, x = 3 and y = 4. Let the line L be perpendicular to 3x + y + 6 = 0 and divide the area of the rectangle into two equal parts. Then the distance of the point ( 12, -5 ) from the line L is equal to :

1. 10
2. 2 5
3. 2 10
4. 3 10

Let A(1, 2) and C(-3, -6) be two diagonally opposite vertices of a rhombus, whose sides AD and BC are parallel to the line 7x - y = 14. If B( , ) and D( , ) are the other two vertices, then | + + + | is equal to :

1. 3
2. 6
3. 1
4. 9

Among the statements (S1) : If A(5, -1) and B(-2, 3) are two vertices of a triangle, whose orthocentre is (0, 0), then its third vertex is (-4, -7) and (S2) : If positive numbers 2a, b, c are three consecutive terms of an A.P., then the lines ax + by + c = 0 are concurrent at (2, -2),

1. both are incorrect
2. only (S2) is correct
3. both are correct
4. only (S1) is correct

Let a point A lie between the parallel lines L1 and L2 such that its distances from L1 and L2 are 6 and 3 units, respectively. Then the area (in sq. units) of the equilateral triangle ABC, where the points B and C lie on the lines L1 and L2, respectively, is :

1. 21 3
2. 12 2
3. 15 6
4. 27

Let a be the length of a side of a square OABC with O being the origin. Its side OA makes an acute angle with the positive x-axis and the equations of its diagonals are ( 3+1)x + ( 3-1)y = 0 and ( 3-1)x - ( 3+1)y + 8 3 = 0. Then a2 is equal to :

1. 48
2. 16
3. 24
4. 32

A line passing through the point P(a, 0) makes an acute angle with the positive x-axis. Let this line be rotated about the point P through an angle 2 in the clockwise direction. If in the new position, the slope of the line is 2 - 3 and its distance from the origin is 1 2, then the value of 3a2 2 - 2 3 is :

1. 8
2. 4
3. 5
4. 6

Let ABC be the triangle such that the equations of lines AB and AC be 3y - x = 2 and x + y = 2, respectively, and the points B and C lie on x-axis. If P is the orthocentre of the triangle ABC, then the area of the triangle PBC is equal to

1. 8
2. 4
3. 10
4. 6

If the orthocenter of the triangle formed by the lines y = x + 1, y = 4x - 8 and y = mx + c is at (3, -1), then m - c is :

1. 0
2. 2
3. -2
4. 4

Let the three sides of a triangle lie on the lines 4x - 7y + 10 = 0, x + y = 5 and 7x + 4y = 15. Then the distance of its orthocentre from the orthocentre of the triangle formed by the lines x = 0, y = 0 and x + y = 1 is

1. 20
2. 20
3. 5
4. 5

A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines L1: 2x+y+6=0 and L2: 4x+2y-p=0, p>0, at the points A and B, respectively. If AB = 9 2 and the foot of the perpendicular from the point A on the line L2 is M, then AMBM is equal to

1. 5
2. 3
3. 2
4. 4

Consider the lines x(3 + 1) + y(7 + 2) = 17 + 5, being a parameter, all passing through a point P. One of these lines (say L) is farthest from the origin. If the distance of L from the point (3, 6) is d, then the value of d2 is

1. 10
2. 20
3. 15
4. 30

Let the area of the triangle formed by a straight line L : x + by + c = 0 with co-ordinate axes be 48 square units. If the perpendicular drawn from the origin to the line L makes an angle of 45 with the positive x-axis, then the value of b2 + c2 is :

1. 90
2. 83
3. 93
4. 97

Let be a triangle formed by the lines 7x - 6y + 3 = 0, x + 2y - 31 = 0 and 9x - 2y - 19 = 0. Let the point (h, k) be the image of the centroid of in the line 3x + 6y - 53 = 0. Then h2 + k2 + hk is equal to :

1. 47
2. 37
3. 40
4. 36

Let the line x + y = 1 meet the axes of x and y at A and B, respectively. A right angled triangle AMN is inscribed in the triangle OAB, where O is the origin and the points M and N lie on the lines OB and AB, respectively. If the area of the triangle AMN is 49 of the area of the triangle OAB and AN : NB = : 1, then the sum of all possible value(s) of is:

1. 12
2. 52
3. 2
4. 136

Two equal sides of an isosceles triangle are along -x + 2y = 4 and x + y = 4. If m is the slope of its third side, then the sum of all possible distinct values of m is:

1. -2 10
2. 12
3. -6
4. 6

Let the lines 3x-4y- =0, 8x-11y-33=0, and 2x-3y+ =0 be concurrent. If the image of the point (1,2) in the line 2x-3y+ =0 is ( 5713, -4013 ), then | | is equal to

1. 91
2. 113
3. 101
4. 84

Let the points ( 112, ) lie on or inside the triangle with sides x+y=11, x+2y=16 and 2x+3y=29. Then the product of the smallest and the largest values of is equal to :

1. 22
2. 33
3. 55
4. 44

Let the area of a PQR with vertices P(5,4), Q(-2,4) and R(a,b) be 35 square units. If its orthocenter and centroid are O (2, 145 ) and C(c,d) respectively, then c+2d is equal to

1. 3
2. 73
3. 2
4. 83

A rod of length eight units moves such that its ends A and B always lie on the lines x-y+2=0 and y+2=0, respectively. If the locus of the point P, that divides the rod AB internally in the ratio 2:1 is 9 (x2+ y2+ xy+ x+28y )-76=0, then - - is equal to :

1. 24
2. 22
3. 21
4. 23

Let the triangle PQR be the image of the triangle with vertices (1,3),(3,1) and (2,4) in the line x+2y=2. If the centroid of is the point ( , ), then 15( - ) is equal to :

1. 21
2. 19
3. 22
4. 24

A variable line L passes through the point (3, 5) and intersects the positive coordinate axes at the points A and B. The minimum area of the triangle OAB, where O is the origin, is :

1. 35
2. 25
3. 30
4. 40

A ray of light coming from the point P(1, 2) gets reflected from the point Q on the x-axis and then passes through the point R(4, 3). If the point S(h, k) is such that PQRS is a parallelogram, then hk2 is equal to:

1. 60
2. 70
3. 80
4. 90