The equation of a circle with origin as centre and passing through the vertices of an equilateral triangle whose median is of length is:

Explanation: 1. Identify the Relationship Between Median and Radius For an equilateral triangle, the centroid ( ) is the same as the circumcenter ( ). The origin is given as the center of the circle, so the origin is the circumcenter. In an equilateral triangle, the centroid divides the median in the ratio . The distance from the vertex to the centroid is the radius ( ) of the circumcircle. Given the length of the median is : 2. Equation of the Circle The standard equation of a circle with center at the origin and radius is: Substitute into the equation: Final Result The equation of the circle is:

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