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Integration of Irrational Algebraic Fractions - 1

If A1, A2.....

Question

If $A_{1}, A_{2} . . . . . . . . A_{n}$ are the vertices of a regular plane polygon with $n$ sides and $O$ is the center then $n - 1 \sum i = 1 (\to O A_{i} \times \to O A_{i + 1}) =$

(n + 1) ({\to O A}_{1} \times {\to O A}_{2})

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(n + 1) ({\to O A}_{2} \times {\to O A}_{1})

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(n - 1)

({\to O A}_{1} \times {\to O A}_{2})

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(n - 1)

({\to O A}_{2} \times {\to O A}_{1})

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Solution

The correct option is B $(n - 1)$ $({\to O A}_{1} \times {\to O A}_{2})$
$\sum_{i = 1}^{n - 1} (_{i} \times_{i + 1})$
$_{1} \times_{2} = |_{1} | |_{2} | sin θ^n$
$_{1},_{2}, . . . . . ._{n}$ will be of same length and subtend same angle at the centre.
$∴ \sum_{i = 1}^{n - 1} (_{i} \times_{i + 1}) = (n - 1) |_{1} | |_{2} | sin θ^n$
$∴ \sum_{i = 1}^{n - 1} (_{i} \times_{i + 1}) = (n - 1) (_{1} \times_{2})$
Hence, option C is correct.

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