The circumcentre of the triangle formed by the points $A(1, \sqrt{3}), B(-1,-\sqrt{3})$ and $(3,-\sqrt{3})$ is
The circumcentre of the triangle formed by the points $A(1, \sqrt{3}), B(-1,-\sqrt{3})$ and $(3,-\sqrt{3})$ is
$(1,-\sqrt{3})$
$\left(-1, \frac{1}{\sqrt{3}}\right)$
$(0,0)$
$\left(1, \frac{-1}{\sqrt{3}}\right)$
Solution
Vertices of $\triangle A B C$ are
$A(1, \sqrt{3}), B(-1,-\sqrt{3})$ and $C(3,-\sqrt{3})$
$A B=\sqrt{4+12}=4$
$\begin{aligned} & B C=\sqrt{4+12}=4 \\ & A C=\sqrt{4+12}=4\end{aligned}$
Here, $A B C$ is an equilateral triangle.
$\therefore$ Circumcentre of triangle is
$\left(\frac{1-1+3}{3}, \frac{\sqrt{3}-\sqrt{3}-\sqrt{3}}{3}\right)$
$=\left(1, \frac{-1}{\sqrt{3}}\right)$ $[\because$ in equilateral triangle circumcentre and centroid coincide]