Find the condition that the st. lines : `p_1x+q_1y=1,p_2x+q_2y=1 and p_3x+q_3y=1` be concurrent, show that the point `(p_1,q_1),(p_2,q_2)` and `(p_3,q_3)` are collinear.

Show that the points P(2,6),Q(1,2)and R(3,10) are collinear.

Find p if the points (p+1,1), (2p + 1,3) and (2p + 2, 2p) are collinear.

Find the product: (p+3q) (3p-q)

Find the ratio in which the plane ax+by+cz+d=0 divides the join of the points P(x_1,y_1,z_1) and Q(x_2,y_2,z_2) lying on the plane. Hence, show that the points P(1,-2,3) and Q(0,0,-1) lie on opposite sides of the plane 2x+5y+7z=3.

If the three points (3q,0), (0, 3p) and (1, 1) are collinear, then which one is true ?

Let S_p and S_q be the coefficients of x^p and x^q respectively in (1 + x)^(p+q) , then :

A=[{:(l_(1),m_(1),n_(1)),(l_(2),m_(2),n_(2)),(l_(3),m_(3),n_(3)):}] and B=[{:(p_(1),q_(1),r_(1)),(p_(2),q_(2),r_(2)),(p_(3),q_(3),r_(3)):}] Where p_(i), q_(i),r_(i) are the co-factors of the elements l_(i), m_(i), n_(i) for i=1,2,3 . If (l_(1),m_(1),n_(1)),(l_(2),m_(2),n_(2)) and (l_(3),m_(3),n_(3)) are the direction cosines of three mutually perpendicular lines then (p_(1),q_(1), r_(1)),(p_(2),q_(2),r_(2)) and (p_(3),q_(),r_(3)) are

Find the direction cosines of the ray form P to Q where P is point (1,-2,2) and Q is te point (3,-5,-4).

Find the points on the line segment PQ situated at 2/3 of the distance from the initial point P to the final point Q, where P=(3,1,7),Q=(-2,5,3) .