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`

Show that two lines a_(1)x + b_(1) y+ c_(1) = 0 " and " a_(2)x + b_(2) y + c_(2) = 0 " where " b_(1) , b_(2) ne 0 are : (i) Parallel if a_(1)/b_(1) = a_(2)/b_(2) , and (ii) Perpendicular if a_(1) a_(2) + b_(1) b_(2) = 0 .

If a_1x^3 + b_1x² + c_1x + d_1 = 0 and a_2x^3 + b_2x^2+ c_2x + d_2 = 0 have a pair of repeated common roots, then prove that |[3a_1,2b_1,c_1],[3a_2,2b_2,c_2],[a_2b_1-a_1b_2,c_1a_2-c_2a_1,d_1a_2-d_2a_1]|=0

The two equations a_1x+b_1y=k_1 and a_2x+b_2y=k_2 will have unique solutions for x and y when

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.

Consider two lines L_1a n dL_2 given by a_1x+b_1y+c_1=0a n da_2x+b_2y+c_2=0 respectively where c1 and c2 !=0, intersecting at point PdotA line L_3 is drawn through the origin meeting the lines L_1a n dL_2 at Aa n dB , respectively, such that PA=P B . Similarly, one more line L_4 is drawn through the origin meeting the lines L_1a n dL_2 at A_1a n dB_2, respectively, such that P A_1=P B_1dot Obtain the combined equation of lines L_3a n dL_4dot

If the circles x^(2) + y^(2) + 2ax + c^(2) = 0 and x^(2) + y^(2) + 2by + c^(2) = 0 touch each other, prove that, (1)/(a^(2)) + (1)/(b^(2)) = (1)/(c^(2)) .

If the simultaneous linear equations a_(1)x+b_(1)y+c_(1)=0 " and " a_(2)x+b_(2)y+c_(2)=0 have only one solution, then the required condition is -

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 the lines x=a_(1)y+b_(1),z=c_(1)y+d_(1) and x=a_(2)y+b_(2),z=c_(2)y+d_(2) are perpendicular, prove that, 1+a_(1)a_(2)+c_(1)c_(2)=0 .