If the lines `p_1x+q_1y=1,p_2x+q_2y=1a n dp_3x+q_3y=1,` be concurrent, show that the point `(p_1, q_1),(p_2, q_2)a n d(p_3, q_3)` are collinear.

Show that the points P (-2, 3, 5) , Q (1, 2, 3) and R (7, 0, -1) are collinear.

If veca,vecb be two non zero non parallel vectors then show that the points whose position vectors are p_1veca+q_1vecb,p_2veca+q_2vecb,p_3veca+q_3vecb are collinear if |(1,p_1,q_1),(1,p_2,q_2),(1,p_3,q_3)|=0

If each pair of the three equations x^(2) - p_(1)x + q_(1) =0, x^(2) -p_(2)c + q_(2)=0, x^(2)-p_(3)x + q_(3)=0 have common root, prove that, p_(1)^(2)+ p_(2)^(2) + p_(3)^(2) + 4(q_(1)+q_(2)+q_(3)) =2(p_(2)p_(3) + p_(3)p_(1) + p_(1)p_(2))

the lines (p+2q)x+(p-3q)y=p-q for different values of p&q passes trough the fixed point is:

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

If (x_1, y_1)&(x_2,y_2) are the ends of a diameter of a circle such that x_1&x_2 are the roots of the equation a x^2+b x+c=0 and y_1&y_2 are the roots of the equation p y^2=q y+c=0. Then the coordinates of the centre of the circle is: (b/(2a), q/(2p)) ( b/(2a), q/(2p)) (b/a , q/p) d. none of these

If the line of regression of Y on X is p_(1)x + q_(1)y + r_(1)= 0 and that of X on Y is p_(2)x + q_(2)y + r_(2)= 0 , prove that (p_(1)q_(2))/(p_(2)q_(1)) le 1

The point of injtersection of the line x/p+y/q=1 and x/q+y/p=1 lies on the line

If x ,y \ a n d \ z are not all zero and connected by the equations a_1x+b_1y+c_1z=0,a_2x+b_2y+c_2z=0 , and (p_1+lambdaq_1)x+(p_2+lambdaq_2)+(p_3+lambdaq_3)z=0 , show that lambda=-|[a_1,b_1,c_1],[a_2,b_2,c_2],[p_1,p_2,p_3]|-:|[a_1,b_1,c_1],[a_2,b_2,c_2],[q_1,q_2,q_3]|

Show that |(1, 1+p, 1+p+q), (2, 3+2p, 4+3p+2q), (3, 6+3p, 10+6p+3q)|=1.