Let A_0 A_1 A_2 A_3 A_4 A_5 be a regular hexa | vedclass

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(C) In a regular hexagon inscribed in a circle of radius $R = 1$,the side length is equal to the radius,so $A_0A_1 = A_1A_2 = 1$.The segment $A_0A_4$

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$\sqrt{3}$

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(C) In a regular hexagon inscribed in a circle of radius $R = 1$,the side length is equal to the radius,so $A_0A_1 = A_1A_2 = 1$.The segment $A_0A_4$ is a chord of the circle. In a regular hexagon,the angle subtended by a side at the center is $60^\circ$. The segment $A_0A_4$ subtends an angle of $120^\circ$ at the center (or $240^\circ$ depending on the arc).Alternatively,using the geometry of the hexagon,$A_0A_4$ is the distance between two vertices separated by one vertex $(A_5)$. In a circle of radius $R$,the length of a chord subtending an angle $	heta$ at the center is $2R \sin(	heta/2)$.For $A_0A_4$,the angle subtended at the center is $120^\circ$,so $A_0A_4 = 2(1) \sin(120^\circ/2) = 2 \sin(60^\circ) = 2 	imes \frac{\sqrt{3}}{2} = \sqrt{3}$.The product of the lengths $A_0A_1$,$A_1A_2$,and $A_0A_4$ is $(1) 	imes (1) 	imes (\sqrt{3}) = \sqrt{3}$.

Column $I$	Column $II$
$(A)$ Two intersecting circles	$(p)$ have a common tangent
$(B)$ Two mutually external circles	$(q)$ have a common normal
$(C)$ Two circles,one strictly inside the other	$(r)$ do not have a common tangent
$(D)$ Two branches of a hyperbola	$(s)$ do not have a common normal

Let $A_0 A_1 A_2 A_3 A_4 A_5$ be a regular hexagon inscribed in a circle of unit radius. The product of the lengths of the line segments $A_0A_1$,$A_1A_2$,and $A_0A_4$ is

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