Straight Lines

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

If P is a point on the circle x2 + y2 = 4, Q is a point on the straight line 5x + y + 2 = 0 and x - y + 1 = 0 is the perpendicular bisector of PQ, then 13 times the sum of abscissa of all such points P is .

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