Let $$\alpha$$_{1}, $$\alpha$$_{2} ($$\alpha$$_{1} < $$\alpha$$_{2}) be the values of $$\alpha$$ fo the points ($$\alpha$$, $$-$$3), (2, 0) and (1, $$\alpha$$) to be collinear. Then the equation of the line, passing through ($$\alpha$$_{1}, $$\alpha$$_{2}) and making an angle of $${\pi \over 3}$$ with the positive direction of the x-axis, is :

The distance of the origin from the centroid of the triangle whose two sides have the equations $$x - 2y + 1 = 0$$ and $$2x - y - 1 = 0$$ and whose orthocenter is $$\left( {{7 \over 3},{7 \over 3}} \right)$$ is :

The distance between the two points A and A' which lie on y = 2 such that both the line segments AB and A' B (where B is the point (2, 3)) subtend angle $${\pi \over 4}$$ at the origin, is equal to :

Let a triangle be bounded by the lines L_{1} : 2x + 5y = 10; L_{2} : $$-$$4x + 3y = 12 and the line L_{3}, which passes through the point P(2, 3), intersects L_{2} at A and L_{1} at B. If the point P divides the line-segment AB, internally in the ratio 1 : 3, then the area of the triangle is equal to :