Straight Lines

A rod of length eight units moves such that its ends A and B always lie on the lines x-y+2=0 and y+2=0, respectively. If the locus of the point P, that divides the rod AB internally in the ratio 2:1 is 9 (x2+ y2+ xy+ x+28y )-76=0, then - - is equal to :

1. 24
2. 22
3. 21
4. 23

Let the triangle PQR be the image of the triangle with vertices (1,3),(3,1) and (2,4) in the line x+2y=2. If the centroid of is the point ( , ), then 15( - ) is equal to :

1. 21
2. 19
3. 22
4. 24

Let A(6,8), B(10 ,-10 ) and C(-10 ,10 ), be the vertices of a triangle. If L(a,9) and G(h,k) be its orthocenter and centroid respectively, then (5a-3h+6k+100 2 ) is equal to .

Let the distance between two parallel lines be 5 units and a point P lie between the lines at a unit distance from one of them. An equilateral triangle PQR is formed such that Q lies on one of the parallel lines, while R lies on the other. Then (QR)2 is equal to .

A variable line L passes through the point (3, 5) and intersects the positive coordinate axes at the points A and B. The minimum area of the triangle OAB, where O is the origin, is :

1. 35
2. 25
3. 30
4. 40

A ray of light coming from the point P(1, 2) gets reflected from the point Q on the x-axis and then passes through the point R(4, 3). If the point S(h, k) is such that PQRS is a parallelogram, then hk2 is equal to:

1. 60
2. 70
3. 80
4. 90

Two vertices of a triangle ABC are A(3, -1) and B(-2, 3), and its orthocentre is P(1, 1). If the coordinates of the point C are ( , ) and the centre of the circle circumscribing the triangle PAB is (h, k), then the value of ( + ) + 2(h + k) equals

1. 81
2. 15
3. 51
4. 5

The equations of two sides AB and AC of a triangle ABC are 4x + y = 14 and 3x - 2y = 5, respectively. The point (2, - 43 ) divides the third side BC internally in the ratio 2 : 1, the equation of the side BC is

1. x + 6y + 6 = 0
2. x - 3y - 6 = 0
3. x + 3y + 2 = 0
4. x - 6y - 10 = 0

If the orthocentre of the triangle formed by the lines 2x + 3y - 1 = 0, x + 2y - 1 = 0 and ax + by - 1 = 0, is the centroid of another triangle, whose circumcentre and orthocentre respectively are (3, 4) and (-6, -8), then the value of |a - b| is .

If the line segment joining the points (5, 2) and (2, a) subtends an angle 4 at the origin, then the absolute value of the product of all possible values of a is :

1. 4
2. 8
3. 6
4. 2

If the image of the point (-4, 5) in the line x + 2y = 2 lies on the circle (x + 4)2 + (y - 3)2 = r2, then r is equal to:

1. 2
2. 3
3. 4
4. 1

Let a ray of light passing through the point (3, 10) reflects on the line 2x + y = 6 and the reflected ray passes through the point (7, 2). If the equation of the incident ray is ax + by + 1 = 0, then a2 + b2 + 3ab is equal to .

Let a variable line of slope m > 0 passing through the point (4, -9) intersect the coordinate axes at the points A and B. The minimum value of the sum of the distances of A and B from the origin is

1. 30
2. 15
3. 10
4. 25

If P(6, 1) be the orthocentre of the triangle whose vertices are A(5, -2), B(8, 3) and C(h, k), then the point C lies on the circle :

1. x2 + y2 - 74 = 0
2. x2 + y2 - 65 = 0
3. x2 + y2 - 61 = 0
4. x2 + y2 - 52 = 0

If the locus of the point, whose distances from the point (2, 1) and (1, 3) are in the ratio 5 : 4, is ax2 + by2 + cxy + dx + ey + 170 = 0, then the value of a2 + 2b + 3c + 4d + e is equal to :

1. 37
2. -27
3. 437
4. 5

Let two straight lines drawn from the origin O intersect the line 3x + 4y = 12 at the points P and Q such that OPQ is an isosceles triangle and POQ = 90 . If l = OP2 + PQ2 + QO2, then the greatest integer less than or equal to l is :

1. 42
2. 46
3. 48
4. 44

Let A(-1, 1) and B(2, 3) be two points and P be a variable point above the line AB such that the area of PAB is 10. If the locus of P is ax + by = 15, then 5a + 2 b is :

1. - 125
2. - 65
3. 6
4. 4

The vertices of a triangle are A(-1, 3), B(-2, 2) and C(3,-1). A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is :

1. -x+y-(2- 2)=0
2. x+y-(2- 2)=0
3. x+y+(2- 2)=0
4. x-y-(2+ 2)=0

Consider a triangle ABC having the vertices A(1, 2), B( , ) and C( , ) and angles = 6 and = 2 3. If the points B and C lie on the line y = x + 4, then 2 + 2 is equal to .

Let ABC be an isosceles triangle in which A is at (-1,0), A= 2 3,AB=AC and B is on the positive x-axis. If BC=4 3 and the line BC intersects the line y=x+3 at ( , ), then 4 2 is .

Let , , , Z and let A( , ), B(1,0), C( , ) and D(1,2) be the vertices of a parallelogram ABCD. If AB= 10 and the points A and C lie on the line 3y=2x+1, then 2( + + + ) is equal to

1. 8
2. 5
3. 12
4. 10

Let a variable line passing through the centre of the circle x2+y2-16x-4y=0, meet the positive co-ordinate axes at the points A and B. Then the minimum value of OA + OB, where O is the origin, is equal to

1. 12
2. 20
3. 24
4. 18

Let A(a, b), B(3, 4) and C(-6,-8) respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point P(2a+3, 7b+5) from the line 2x+3y-4=0 measured parallel to the line x-2y-1=0 is

1. 17 56
2. 15 57
3. 17 57
4. 517

Let A(-2,-1), B(1,0), C( , ) and D( , ) be the vertices of a parallelogram ABCD. If the point C lies on 2x-y=5 and the point D lies on 3x-2y=6, then the value of | + + + | is equal to .

A line passing through the point A(9,0) makes an angle of 30 with the positive direction of x-axis. If this line is rotated about A through an angle of 15 in the clockwise direction, then its equation in the new position is :

1. y 3+2+x=9
2. x 3+2+y=9
3. x 3-2+y=9
4. y 3-2+x=9

If x2-y2+2hxy+2gx+2fy+c=0 is the locus of a point, which moves such that it is always equidistant from the lines x+2y+7=0 and 2x-y+8=0, then the value of g+c+h-f equals

1. 8
2. 14
3. 29
4. 6

Let (5, a4 ) be the circumcenter of a triangle with vertices A(a,-2), B(a, 6) and C ( a4,-2 ). Let denote the circumradius, denote the area and denote the perimeter of the triangle. Then + + is

1. 60
2. 62
3. 53
4. 30

In a ABC, suppose y=x is the equation of the bisector of the angle B and the equation of the side AC is 2x-y=2. If 2AB=BC and the points A and B are respectively (4,6) and ( , ), then +2 is equal to

1. 42
2. 39
3. 48
4. 45

Let A be the point of intersection of the lines 3x+2y=14, 5x-y=6 and B be the point of intersection of the lines 4x+3y=8, 6x+y=5. The distance of the point P(5,-2) from the line AB is

1. 132
2. 8
3. 52
4. 6

The distance of the point (2,3) from the line 2x-3y+28=0, measured parallel to the line 3x-y+1=0, is equal to

1. 3+4 2
2. 6 3
3. 4+6 3
4. 4 2