Question

Write the area of the figure formed by the lines a |x| + b |y| + c = 0.

 

Answer 1

The given lines can be written separately in the following way:
a x + b y + c = 0;  x, y \[\geq\] 0         ... (1)

\[-\] a x + b y + c = 0;  x < 0 y  \[\geq\]0              ... (2)

\[-\] a x \[-\] b y + c = 0;  x < 0 y < 0             ... (3) 

a x \[-\] b y + c = 0;  x  \[\geq\] 0 y < 0             ... (4)

The lines and the region enclosed between them is shown below.

So, the area of the figures formed by the lines a |x| + b |y| + c = 0 is \[4 \times \frac{1}{2}\left| \frac{c}{a} \times \frac{c}{b} \right| = \frac{2 c^2}{\left| ab \right|}\] square units.