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

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

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 .

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 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

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