Straight Lines · Question 5

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

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

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 .