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.

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:

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 .

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 .

The orthocentre of the triangle formed by the lines x y=0 and x+y=1 is (a) (1/2,1/2) (b) (1/3,1/3) (c) (0,0) (d) (1/4,1/4)

The area of the triangle formed by the lines y=a x ,x+y-a=0 and the y-axis is (a) 1/(2|1+a|) (b) 1/(|1+a|) (c) 1/2|a/(1+a)| (d) (a^2)/(2|1+a|)

Show that the circle passing through the origin and cutting the circles x^2+y^2-2a_1x-2b_1y+c_1=0 and x^2+y^2-2a_2x -2b_2y+c_2=0 orthogonally is |(x^2+y^2,x,y),(c_1,a_1,b_1),(c_2,a_2,b_2)|=0

Find the equation of the line which passes thorugh the point P(alpha, beta, gamma) and is parallel to the line a_1x+b_1y+c_1z+d_1=0, a_2x+b_2y+c_2z+d_2=0

findthe equationof the plane passing through the point (alpha, beta, gamma) and perpendicular to the planes a_1x+b_1y+c_1z+d_1=0 and a_2x+b_2y+c_2z+d_2=0

If the lines a_1x+b_1y+1=0,a_2x+b_2y+1=0 and a_3x+b_3y+1=0 are concurrent, show that the point (a_1, b_1),(a_1, b_2) and (a_3, b_3) are collinear.

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]|