A circle circumscribing an equilateral triangle with centroid at `(0,0)` of a side a isdrawn and a square is drawn touching its four sides to circle. The equation ofcircle circumscribing the square is :

Options

A. `x^(2)+y^(2)=2a^(2)`
B. `3x^(2)+3y^(2)=2a^(2)`
C. `5x^(2)+5y^(2)=3a^(2)`
D. none of these

Explanation

To find the equation of the circle that circumscribes a square which touches the sides of a circle circumscribing an equilateral triangle with a centroid at (0,0) and side length \( a \), we can follow these steps:

### Step 1: Determine the Circumradius of the Equilateral Triangle
The circumradius \( R \) of an equilateral triangle with side length \( a \) is given by the formula:
\[ R = \frac{a}{\sqrt{3}} \]

### Step 2: Locate the Centroid and Vertices of the Triangle
Given that the centroid of the triangle is at the origin (0,0), we can denote the vertices of the triangle as \( A \), \( B \), and \( C \). The coordinates of these vertices can be calculated as follows:
- Vertex \( A \): \( \left(-\frac{a}{2}, -\frac{a \sqrt{3}}{6}\right) \)
- Vertex \( B \): \( \left(\frac{a}{2}, -\frac{a \sqrt{3}}{6}\right) \)
- Vertex \( C \): \( \left(0, \frac{a \sqrt{3}}{3}\right) \)

### Step 3: Determine the Radius of the Circle Circumscribing the Square
The square is inscribed in the circle that circumscribes the triangle. The radius of the circle that circumscribes the square is equal to the circumradius of the triangle, which we found to be \( R = \frac{a}{\sqrt{3}} \).

### Step 4: Find the Radius of the Circle Circumscribing the Square
The radius of the circle that circumscribes the square can be calculated using the relationship between the side length of the square \( s \) and the circumradius \( R \):
\[ R_{square} = \frac{s \sqrt{2}}{2} \]
Since the square touches the circle, we have:
\[ R_{square} = R = \frac{a}{\sqrt{3}} \]
Thus, we can equate:
\[ \frac{s \sqrt{2}}{2} = \frac{a}{\sqrt{3}} \]
From this, we can solve for \( s \):
\[ s = \frac{2a}{\sqrt{6}} = \frac{a \sqrt{6}}{3} \]

### Step 5: Determine the Radius of the Circle Circumscribing the Square
The radius of the circle that circumscribes the square is the same as the circumradius of the square:
\[ R_{circumscribing\ circle} = \frac{s \sqrt{2}}{2} = \frac{\left(\frac{a \sqrt{6}}{3}\right) \sqrt{2}}{2} = \frac{a \sqrt{12}}{6} = \frac{a \sqrt{3}}{3} \]

### Step 6: Write the Equation of the Circle
The equation of a circle with center at the origin and radius \( R \) is given by:
\[ x^2 + y^2 = R^2 \]
Substituting \( R = \frac{a \sqrt{3}}{3} \):
\[ x^2 + y^2 = \left(\frac{a \sqrt{3}}{3}\right)^2 = \frac{3a^2}{9} = \frac{a^2}{3} \]

### Step 7: Final Equation of the Circle
To express the equation in standard form:
\[ 3x^2 + 3y^2 = a^2 \]

Thus, the equation of the circle that circumscribes the square is:
\[ \boxed{3x^2 + 3y^2 = a^2} \]