If the lines `p_1 x + q_1 y = 1, p_2 x + q_2 y=1 and p_3 x + q_3 y = 1` be concurrent, show that the points `(p_1 , q_1), (p_2 , q_2 ) and (p_3 , q_3)` are colliner.

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.

If the lines p_(1)x+q_(1)y=1+q_(2)y=1 and p_(3)x+q_(3)y=1 be concurrent,show that the point (p_(1),q_(1)),(p_(2),q_(2)) and (p_(3),q_(3)) 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

Let C_1 and C_2 be parabolas x^2 = y - 1 and y^2 = x-1 respectively. Let P be any point on C_1 and Q be any point C_2. Let P_1 and Q_1 be the reflection of P and Q, respectively w.r.t the line y = x then prove that P_1 lies on C_2 and Q_1 lies on C_1 and PQ >= [P P_1, Q Q_1]. Hence or otherwise , determine points P_0 and Q_0 on the parabolas C_1 and C_2 respectively such that P_0 Q_0 <= PQ for all pairs of points (P,Q) with P on C_1 and Q on C_2