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 .

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 .

Consider a system of linear equation in three variables x,y,z a_1x+b_1y+ c_1z = d_1 , a_2x+ b_2y+c_2z=d_2 , a_3x + b_3y + c_3z=d_3 The systems can be expressed by matrix equation [(a_1,b_1,c_1),(a_2,b_2,c_2),(c_1,c_2,c_3)][(x),(y),(z)]=[(d_1),(d_2),(d_3)] if A is non-singular matrix then the solution of above system can be found by X = A^(-1)B , the solution in this case is unique. if A is a singular matrix i.e. then the system will have no unique solution if no solution (i.e. it is inconsistent) if Where Adj A is the adjoint of the matrix A, which is obtained by taking transpose of the matrix obtained by replacing each element of matrix A with corresponding cofactors. Now consider the following matrix. A=[(a,1,0),(1,b,d),(1,b,c)], B=[(a,1,1),(0,d,c),(f,g,h)], U=[(f),(g),(h)], V=[(a^2),(0),(0)], X=[(x),(y),(z)] If AX=U has infinitely many solutions then the equation BX=U is consistent if

Consider a system of linear equation in three variables x,y,z a_1x+b_1y+ c_1z = d_1 , a_2x+ b_2y+c_2z=d_2 , a_3x + b_3y + c_3z=d_3 The systems can be expressed by matrix equation [(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)][(x),(y),(z)]=[(d_1),(d_2),(d_3)] if A is non-singular matrix then the solution of above system can be found by X = A^(-1)B , the solution in this case is unique. if A is a singular matrix i.e. then the system will have no solution (i.e. it is inconsistent) if Where Adj A is the adjoint of the matrix A, which is obtained by taking transpose of the matrix obtained by replacing each element of matrix A with corresponding cofactors. Now consider the following matrix. A=[(a,1,0),(1,b,d),(1,b,c)], B=[(a,1,1),(0,d,c),(f,g,h)], U=[(f),(g),(h)], V=[(a^2),(0),(0)], X=[(x),(y),(z)] The system AX=U has infinitely many solutions if :

Find the equation of the plane which passes through the line of intersection of the planes a_1x+b_1y+c_1z+d_1=0 and a_2x+b_2y+c_2z+d_2=0 and which is parallel to the line (x-alpha)/l=(y-beta)/m=(z-gamma)/n

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)

If a_1x^2 + b_1 x + c_1 = 0 and a_2x^2 + b_2 x + c_2 = 0 has a common root, then the common root is

Two conics a_1x^2+2h_1xy + b_1y^2 = c_1, a_2x^2 + 2h_2xy+b_2y^2 = c_2 intersect in 4 concyclic points. Then

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]|