Straight Lines

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

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

Let the lines 3x-4y- =0, 8x-11y-33=0, and 2x-3y+ =0 be concurrent. If the image of the point (1,2) in the line 2x-3y+ =0 is ( 5713, -4013 ), then | | is equal to

1. 91
2. 113
3. 101
4. 84

Let the points ( 112, ) lie on or inside the triangle with sides x+y=11, x+2y=16 and 2x+3y=29. Then the product of the smallest and the largest values of is equal to :

1. 22
2. 33
3. 55
4. 44

Let the area of a PQR with vertices P(5,4), Q(-2,4) and R(a,b) be 35 square units. If its orthocenter and centroid are O (2, 145 ) and C(c,d) respectively, then c+2d is equal to

1. 3
2. 73
3. 2
4. 83

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 .