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 parallelogram formed by the lines x/a+y/b=1,x/b+y/a=1,x/a+y/b=2and x/b+y/a=2 is a rhombus.

Represent the following equations in matrix form: a_(1)x+b_(1)y+c_(1)=0 a_(2)x+b_(2)y+c_(2)=0

The straight line a_(1)x+b_(1)y+c_(1)=0 anda_(2)x+b_(2)y+c_(2)=0 are parallel to each other if -

Find the condition so that the two equations a_1 x^2+b_1 x+c_1=0 and a_2 x^2+b_2 x+c_2=0 will have a common root.

Statement - I: Equation of bisectors of the angles between the liens x=0 and y=0 are y=+-x Statement - II : Equation of the bisectors of the angles between the lines a_(1)x+b_(1)y+c_(1)=0anda_(2)x+b_(2)y+c_(2)=0 are (a_(1)x+b_(1)y+c_(1))/(sqrt(a_(1)^(2)+b_(1)^(2)))=+-(a_(2)x+b_(2)y+c_(2))/(sqrt(a_(2)^(2)+b_(2)^(2))) (Provided a_(1)b_(2)nea_(2)b_(1)andc_(1),c_(2)gt0)

Show that the condition for the lines x= a_1z+b_1,y = c_1z+d_1 and x = a_2z+b_2, y = c_2z+ d_2 be perpendicular is a_1a_2+c_1c_2=0

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

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 the equation of the straight line throught (alpha,beta) and through the point of intersection of the lines a_(1)x+b_(1)y+c_(1)=0 anda_(2)x+b_(2)y+c_(2)=0 is (a_(1)x+b_(1)y+c_(1))/(a_(1)alpha+b_(1)beta+c_(1))=(a_(2)x+b_(2)y+c_(2))/(a_(2)alpha+b_(2)beta+c_(2))

If the quadrilateral formed by the lines a x+b y+c=0,a^(prime)x+b^(prime)y+c=0,a x+b y+c^(prime)=0,a^(prime)x+b^(prime)y+c^(prime) =0 has perpendicular diagonals, then (a) b^2+c^2=b^('2)+c^('2) (b) c^2+a^2=c^('2)+a^('2) (c) a^2+b^2=a^('2)+b^('2) (d) none of these