Given that we need to find the equation of circumcircle of an equilateral triangle whose centroid is (1,1) and one of its vertex is (- 1,2).

We know that in an equilateral triangle, the centroid and circumcentre coincides and circumcircle passes through all the vertices of the triangle.

We have a circle with centre (1,1) and passing through the point (- 1,2).

We know that the radius of the circle is the distance between the centre and any point on the radius. So, we find the radius of the circle.

We know that the distance between the two points (x₁,y₁) and (x₂,y₂) is .

Let us assume r is the radius of the circle.

⇒

⇒

⇒

⇒

We know that the equation of the circle with centre (p, q) and having radius ‘r’ is given by:

⇒ (x - p)² + (y - q)² = r²

Now we substitute the corresponding values in the equation:

⇒

⇒ x² - 2x + 1 + y² - 2y + 1 = 5

⇒ x² + y² - 2x - 2y - 3 = 0

∴The correct option is (a).