the diagonals of the parallelogram formed by the the lines `a_1x+b_1y+c_1=0 ,a_1x+b_1y+c_1 '=0 , a_2x+b_2y+c_1=0 , a_2x+b_2y+c_1 '=0` will be right angles if:

Prove that the area of the parallelogram formed by the lines a_1x+b_1y+c_1=0,a_1x+b_1y+d_1=0,a_2x+b_2y+c_2=0, a_2x+b_2y+d_2=0, is |((d_1-c_1)(d_2-c_2))/(a_1b_2-a_2b_1)| sq. units.

Show that the parallelogram formed by ax+by+c=0, a_1 x+ b_1 y+c=0, ax+by+c_1 =0 and a_1 x+b_1 y+c_1 =0 will be a rhombus if a^2 +b^2 = (a_1)^2 + (b_1)^2 .

A : The area of the parallelogram formed by 4x-7y-13=0, 8x-y-39=0, 4x-7y+39=0, 8x-y+13=0 is 52. R : The area of the parallelogram formed by a_(1)x+b_(1)y+c_(1)=0, a_(1)x+b_(1)y+d_(1)=0, a_(2)x+b_(2)y+c_(2)=0, a_(2)x+b_(2)y+d_(2)=0 is |((c_(1)-d_(1))(c_(2)-d_(2)))/(a_(1)b_(2)-a_(2)b_(1))|

If the origin lies in the acute angle between the lines a_1 x + b_1 y + c_1 = 0 and a_2 x + b_2 y + c_2 = 0 , then show that (a_1 a_2 + b_1 b_2) c_1 c_2 lt0 .

If a_1/a_2!=b_1/b_2 then the system of equation a_1x+b_1y+c_1=0 , a_2x+b_2y+c_2=0 has

In the pair of linear equations a_1x+b_1y+c_1=0 and a_2x+ b_2y + C_2 = 0. If a_1/a_2 ne b_1/b_2 then the

a_1x+b_1y+c_1=0 and a_2x+b_2y+c_2=0 has no solution if _________.

a_1x+b_1y+c_1=0" and "a_2x+b_@y+c_2=0 are……..equations.

If the two lines a_1x + b_1y + c_1 = 0 and a_2x + b_2y + c_2 = 0 cut the co-ordinates axes in concyclic points. Prove that a_1a_2 = b_1b_2