SATHEE: Straight Line and Pair of Straight Lines 1 Question 30

Straight Line and Pair of Straight Lines 1 Question 30

30. If $A_{0}, A_{1}, A_{2}, A_{3}, A_{4}$ and $A_{5}$ be a regular hexagon inscribed in a circle of unit radius. Then, the product of the lengths of the line segments $A_{0} A_{1}, A_{0} A_{2}$ and $A_{0} A_{4}$ is

(a) $3 / 4$

(b) $3 \sqrt{3}$

(d) $\frac{3 \sqrt{3}}{2}$

$(1998, 2 M)$

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Solution:

Now, ${(A_{0} A_{1})}^{2} = 1 - {\frac{1}{2}}^{2} + 0 - {\frac{\sqrt{3}}{2}}^{2}$

$\begin{aligned} {(A_{0} A_{2})}^{2} = 1 + {\frac{1}{2}}^{2} + 0 - {\frac{\sqrt{3}}{2}}^{2} \\ = {\frac{3}{2}}^{2} + - {\frac{\sqrt{3}}{2}}^{2} = \frac{9}{4} + \frac{3}{4} = \frac{12}{4} = 3 \\ \Rightarrow A_{0} A_{2} = \sqrt{3} \\ and {(A_{0} A_{4})}^{2} = 1 + {\frac{1}{2}}^{2} + 0 + {\frac{\sqrt{3}}{2}}^{2} \\ = {\frac{3}{2}}^{2} + \frac{3}{4} = \frac{9}{4} + \frac{3}{4} = \frac{12}{4} = 3 \\ \Rightarrow A_{0} A_{4} = \sqrt{3} \\ Thus, (A_{0} A_{1}) (A_{0} A_{2}) (A_{0} A_{4}) = 3 \end{aligned}$

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Straight Line and Pair of Straight Lines 1 Question 30

30. If A0,A1,A2,A3,A4 and A5 be a regular hexagon inscribed in a circle of unit radius. Then, the product of the lengths of the line segments A0A1,A0A2 and A0A4 is

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30. If $A_{0}, A_{1}, A_{2}, A_{3}, A_{4}$ and $A_{5}$ be a regular hexagon inscribed in a circle of unit radius. Then, the product of the lengths of the line segments $A_{0} A_{1}, A_{0} A_{2}$ and $A_{0} A_{4}$ is