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Area of Polygon Using Coordinates

Prove that th...

Question

Prove that the area of the parallelogram formed by the lines $3 x - 4 y + a = 0$ , $3 x - 4 y + 3 a = 0$ , $4 x - 3 y - a = 0$ and $4 x - 3 y - 2 a = 0$ is $\frac{2 a^{2}}{7}$ sq. units.

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Solution

The area of a parallelogram is

$= \frac{| c_{1} - d_{1} | | c_{2} - d_{2} |}{| a_{2} b_{1} - a_{1} b_{2} |}$

$= \frac{| - a + 2 a | | 3 a - a |}{| 3 (- 3) - 4 (- 4) |}$

$= \frac{a \times 2 a}{7}$

$= \frac{2}{7} a^{2}$

Hence proved.

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