If a d-b c 0 , then the area (in sq. units) of the parallelogram formed by the lines a x+b y+2=0, a x+b y+5=0, c x+d y+3=0 and c x+d y+7=0 is

Step-by-step solution

We have, a x+b y+2=0 and a x+b y+5=0 are parallel lines. a x+b y+2=0 y= -a b x- 2 b and a x+b y+5=0 y= -a b x -5 b So, m₁= -a b , c₁= -2 b , c₂=- 5 b Now, again aligned & x+d y+3=0 y= -c d x- 3 d & and c x+d y+7=0 y= -c d x- 7 d aligned So, m₂= -c d , d₁= -3 d , d₂= -7 d Then, area of parallelogram = | (c₁-c₂ ) (d₁-d₂ ) m₁-m₂ | aligned & = | ( -2 b + 5 b ) ( -3 d + 7 d ) - a b + c d | & = | 3 b 4 d -(a d-b c) b d |= | -12 a d-b c |= 12 |a d-b c| . aligned