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:

The line a_(1)x+b_(1)y+c_(1)=0 and a_(2)x+b_(2)y+c_(2)=0 are perpendicular if:

If the lines ax + 2y + 1 = 0, bx + 3y + 1 = 0, cx + 4y + 1 = 0 are concurrent, then a, b, c are in

If the equation of the locus of a point equidistant from the points (a_1, b_1) and (a_2, b_2) is (a_1-a_2)x+(b_1-b_2)y+c=0 , then the value of c is

If the lines a_1x+b_1y+c_1=0 and a_2x+b_2y+c_2=0 cut the coordinae axes at concyclic points, then prove that |a_1a_2|=|b_1b_2|

If x ,ya n dz are not all zero and connected by the equations a_1x+b_1y+c_1z=0,a_2x+b_2y+c_2z=0,a n d(p_1+lambdaq_1)x+(p_2+lambdaq_2)+(p_3+lambdaq_3z=0) , show that lambda=-|a_1b_1c_1a_2b_2c_2p_1p_2p_3|-:|a_1b_1c_1a_2b_2c_2q_1q_2q_3|

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

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

Show that the reflection of the line a x+b y+c=0 on the line x+y+1=0 is the line b+a y+(a+b-c)=0 where a!=bdot

Statement 1: If b c+q r=c a+r p=a b+p q=-1, t h e n|a p a p b q b q c r c r|=0(a b c ,p q r!=0)dot Statement 2: if system of equations a_1x+b_1y+c_1=0,a_2x+b_2y+c_2=0,a_3x+b_2y+c^3=0 has non -trivial solutions |a_1b_1c_1a_2b_2c_2a_3b_3c_3|=0

The number of values of a for which the lines 2x+y-1=0 , a x+3y-3=0, and 3x+2y-2=0 are concurrent is 0 (b) 1 (c) 2 (d) infinite