Straight Lines

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