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