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Question
Area of the triangle formed by the points \[\left( (a + 3)(a + 4), a + 3 \right), \left( (a + 2)(a + 3), (a + 2) \right) \text { and } \left( (a + 1)(a + 2), (a + 1) \right)\]
Options
25a2
5a2
24a2
none of these
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Solution
none of these
The given points are \[(\left\{ a + 3)(a + 4), \left( a + 3 \right) \right\}, \left\{ (a + 2)(a + 3), (a + 2) \right\} \text { and } \left\{ (a + 1)(a + 2), (a + 1) \right\}\].
Let A be the area of the triangle formed by these points.
\[\text { Then, } A = \frac{1}{2}\left[ x_1 \left( y_2 - y_3 \right) + x_2 \left( y_3 - y_1 \right) + x_3 \left( y_1 - y_2 \right) \right]\]
\[ \Rightarrow A = \frac{1}{2}\left[ \left( a + 3 \right)\left( a + 4 \right)\left( a + 2 - a - 1 \right) + \left( a + 2 \right)\left( a + 3 \right)\left( a + 1 - a - 3 \right) + \left( a + 1 \right)\left( a + 2 \right)\left( a + 3 - a - 2 \right) \right]\]
\[ \Rightarrow A = \frac{1}{2}\left[ \left( a + 3 \right)\left( a + 4 \right) - 2\left( a + 2 \right)\left( a + 3 \right) + \left( a + 1 \right)\left( a + 2 \right) \right]\]
\[ \Rightarrow A = \frac{1}{2}\left[ a^2 + 7a + 12 - 2 a^2 - 10a - 12 + a^2 + 3a + 2 \right]\]
\[ \Rightarrow A = 1\]
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