Straight Lines

From the point (-1, -1), two rays are sent making angles of 45 with the line x+y=0. These rays get reflected from the mirror x+2y=1. If the equations of the reflected rays are ax+by=9 and cx+dy=7, a, b, c, d Z, then the value of ad+bc is .

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 A, B be points on the two half-lines x- 3 |y|= , >0 at a distance of from their point of intersection P. The line segment AB meets the angle bisector of the given half-lines at the point Q. If PQ= 92 and R is the radius of the circumcircle of PAB, then 2R is equal to .

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