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Differentiability

Let A1, A2,.....

Question

Let $A_{1}, A_{2}, . . . . A_{n}$ be the vertices of an $n -$ sided regular polygon such that $\frac{1}{A_{1} A_{2}} = \frac{1}{A_{1} A_{3}} + \frac{1}{A_{1} A_{4}}$ . Then the value of $n$ is:

A

7

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B

8

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C

9

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D

10

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Solution

The correct option is A $7$
Let $a$ be the side of $n$ sided regular polygon $A_{1}, A_{2}, A_{3}, A_{4} . . . . A_{n}$
$∴$ Angle subtended by each side at centre $= \frac{2 π}{n}$
Also $O A_{1} = O A_{2} = O A_{3} = . . . . . = O A_{n} = r$ (Say)
In $Δ O A_{1} A_{2},$ $A_{1} A_{2} = 2 r sin \frac{π}{n}$

Similarly in $A_{1} A_{3} = 2 r sin \frac{2 π}{n}$
and in $A_{1} A_{4} = 2 r sin \frac{3 π}{n}$
But given that $\frac{1}{A_{1} A_{2}} = \frac{1}{A_{1} A_{3}} + \frac{1}{A_{1} A_{4}}$
$\Rightarrow \frac{1}{2 sin \frac{π}{n}} = \frac{1}{2 sin \frac{2 π}{n}} + \frac{1}{2 sin \frac{3 π}{n}}$
$\Rightarrow \frac{1}{sin \frac{π}{n}} = \frac{sin \frac{3 π}{n} + sin \frac{2 π}{n}}{sin \frac{2 π}{n} sin \frac{3 π}{n}}$
$\Rightarrow \frac{1}{sin \frac{π}{n}} = \frac{sin \frac{3 π}{n} + sin \frac{2 π}{n}}{2 sin \frac{π}{n} cos \frac{π}{n} sin \frac{3 π}{n}}$
$\Rightarrow 2 cos \frac{π}{n} sin \frac{3 π}{n} = sin \frac{3 π}{n} + sin \frac{2 π}{n}$
$\Rightarrow sin \frac{4 π}{n} + sin \frac{2 π}{n} = sin \frac{3 π}{n} + sin \frac{2 π}{n}$
$∴ sin \frac{4 π}{n} = sin \frac{3 π}{n}$
$\Rightarrow sin \frac{4 π}{n} = sin (π - \frac{3 π}{n})$
$\Rightarrow \frac{4 π}{n} = π - \frac{3 π}{n}$
$\Rightarrow \frac{7 π}{n} = π$
Thus, $n = 7$

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Similar questions

A_{1}, A_{2}, . . . ., A_{n}

be vertices of an n sided regular polygon such that

\frac{1}{A_{1} A_{2}} = \frac{1}{A_{1} A_{3}} + \frac{1}{A_{1} A_{4}}

Q. State true or false:

A_{1}, A_{2}, A_{3} . . . A_{n}

are the vertices of a regular polygon of

n

| O A_{1} | = 1

| A_{1} A_{2} | | A_{1} A_{3} | . . . . | A_{1} A_{n} | = n

A_{1}, A_{2}, \dots, A_{n}

be the vertices of a regular polygon of

n

sides in a circle of radius unity and

a = | A_{1} A_{2} |^{2} + | A_{1} A_{3} |^{2} + \dots | A_{1} A_{n} |^{2},

b = | A_{1} A_{2} | | A_{1} A_{3} | \dots | A_{1} A_{n} |,

\frac{a}{b} =

A_{1} A_{2} A_{3} . . . A_{n}

be a regular polygon of

n

\frac{1}{A_{1} A_{2}} = \frac{1}{A_{1} A_{3}} + \frac{1}{A_{1} A_{4}},

Q. PASSAGE-3:
Let

A_{1}, A_{2}, A_{3} \dots A_{n}

be a regular polygon of

n

sides whose center is origin

O

. Let the complex numbers representing vertices

A_{1}, A_{2}, A_{3} \dots A_{n}

z_{1}, z_{2}, z_{3} \dots z_{n}

respectively. Let

O A_{1} = O A_{2} = \dots = O A_{n} = 1

.

The distances

A_{1} A_{j} (j = 2, 3, \dots, n)

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Differentiability

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