Question

Area of the triangle formed by the points \[\left( (a + 3)(a + 4), a + 3 \right), \left( (a + 2)(a + 3), (a + 2) \right) \text { and } \left( (a + 1)(a + 2), (a + 1) \right)\]

पर्याय

• 25a²

•  5a²

• 24a²

• none of these

Answer 1

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.

\[\text { Then, } A = \frac{1}{2}\left[ x_1 \left( y_2 - y_3 \right) + x_2 \left( y_3 - y_1 \right) + x_3 \left( y_1 - y_2 \right) \right]\]

\[ \Rightarrow A = \frac{1}{2}\left[ \left( a + 3 \right)\left( a + 4 \right)\left( a + 2 - a - 1 \right) + \left( a + 2 \right)\left( a + 3 \right)\left( a + 1 - a - 3 \right) + \left( a + 1 \right)\left( a + 2 \right)\left( a + 3 - a - 2 \right) \right]\]

\[ \Rightarrow A = \frac{1}{2}\left[ \left( a + 3 \right)\left( a + 4 \right) - 2\left( a + 2 \right)\left( a + 3 \right) + \left( a + 1 \right)\left( a + 2 \right) \right]\]

\[ \Rightarrow A = \frac{1}{2}\left[ a^2 + 7a + 12 - 2 a^2 - 10a - 12 + a^2 + 3a + 2 \right]\]

\[ \Rightarrow A = 1\]