Let A1, A2, A3, A4,........, A8 be the vertices of the regular octagons that lie on the circle of radius 2. Let p be a point on the circle and let PAi denote the distance between the point P and Ai for i = 1,2,3,....,8. If P varies over the circle, then the maximum value of the product is PA1.Pa2..........PA8, is
Correct Answer: 512
Solution and Explanation
The vertices \( A_1, A_2, A_3, \dots, A_8 \) form a regular octagon inscribed in a circle with a radius of 2. We need to evaluate \( Z_8 \), which is the result of:
\(Z = (2) \times (1) \times \frac{1}{8}\)
Multiplying the terms:
\(Z_8 = 28 \times 1\)
Thus, the expression \( Z_8 - 28 \) simplifies to:
\(Z_8 - 28 = 28 - 28 = 0\)
Finally, the value of \( \text{MAX}(P_1, P_2, P_8) \) is:
\(2^9 = 512\)
Therefore, the correct answer is \( 2^9 = 512 \).
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Concepts Used:
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