Straight Lines

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