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(A) Let the point of intersection of the two lines be $A$.The angle subtended by the arc $PQ$ at the centre $C$ is twice the angle subtended by the ch

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$180$

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(A) Let the point of intersection of the two lines be $A$.The angle subtended by the arc $PQ$ at the centre $C$ is twice the angle subtended by the chord $PQ$ at any point on the circumference (in the alternate segment).For the line $x + \sqrt{3}y = 1$,the slope is $m_1 = -\frac{1}{\sqrt{3}}$.For the line $\sqrt{3}x - y = 2$,the slope is $m_2 = \sqrt{3}$.Since $m_1 	imes m_2 = (-\frac{1}{\sqrt{3}}) 	imes \sqrt{3} = -1$,the two lines are perpendicular to each other.Therefore,the angle between the lines at their intersection point $A$ is $90^o$.Since the lines are perpendicular,the chord $PQ$ subtends an angle of $90^o$ at the circumference point $A$. By the circle theorem,the angle subtended by the arc $PQ$ at the centre is $2 	imes 90^o = 180^o$.Thus,the chord $PQ$ is a diameter of the circle.

Any circle passes through the point of intersection of the lines $x + \sqrt{3}y = 1$ and $\sqrt{3}x - y = 2$. If it intersects these lines at points $P$ and $Q$,then the angle subtended by the arc $PQ$ at its centre is ............ $^o$.

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