Straight Lines

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 A(4, -2), B(1, 1) and C(9, -3) be the vertices of a triangle ABC. Then the maximum area of the parallelogram AFDE, formed with vertices D, E and F on the sides BC, CA and AB of the triangle ABC respectively, is .

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