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 $\ge$ 0 ... (1)

$-$a x + b y + c = 0; x < 0 y $\ge$ 0 ... (2)

$-$a x $-$ b y + c = 0; x < 0 y < 0 ... (3)

a x $-$ b y + c = 0; x $\ge$ 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

$4\times \frac{1}{2}\left|\frac{c}{a}\times \frac{c}{b}\right|=\frac{2c^2}{\left|ab\right|}$ square units