To solve the problem step by step, we need to find the locus of the centroid of triangle OAB, where O is the origin (0,0), A is on the x-axis, and B is on the y-axis. The circle has a constant radius of 3k and passes through the origin.
### Step-by-Step Solution:
1. **Understanding the Circle**:
The equation of the circle with center (h, k) and radius r is given by:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
Since the circle passes through the origin (0, 0) and has a radius of 3k, we can write the equation of the circle as:
\[
x^2 + y^2 = (3k)^2 = 9k^2
\]
2. **Finding Points A and B**:
Let the coordinates of point A on the x-axis be (a, 0) and the coordinates of point B on the y-axis be (0, b). The points A and B lie on the circle, so they satisfy the circle's equation:
- For point A:
\[
a^2 + 0^2 = 9k^2 \implies a^2 = 9k^2 \implies a = 3k \text{ or } a = -3k
\]
- For point B:
\[
0^2 + b^2 = 9k^2 \implies b^2 = 9k^2 \implies b = 3k \text{ or } b = -3k
\]
3. **Coordinates of Points A and B**:
We can take A as (3k, 0) and B as (0, 3k) without loss of generality.
4. **Finding the Centroid of Triangle OAB**:
The centroid (G) of triangle OAB with vertices O(0,0), A(a,0), and B(0,b) is given by:
\[
G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)
\]
Substituting the coordinates:
\[
G\left(\frac{0 + 3k + 0}{3}, \frac{0 + 0 + 3k}{3}\right) = G(k, k)
\]
5. **Finding the Locus of the Centroid**:
The coordinates of the centroid are (k, k). Since k can vary, we can express k in terms of x and y:
\[
x = k \quad \text{and} \quad y = k
\]
Thus, we have:
\[
x = y
\]
6. **Relating k to the Circle Equation**:
Since the radius of the circle is 3k, we need to relate k to the radius:
\[
k = \frac{x}{3} \quad \text{or} \quad k = \frac{y}{3}
\]
Now substituting k into the circle equation:
\[
x^2 + y^2 = 9k^2 = 9\left(\frac{x}{3}\right)^2 = x^2
\]
Thus, the locus of the centroid can be expressed as:
\[
x^2 + y^2 = 4k^2
\]
### Final Result:
The locus of the centroid of triangle OAB is given by:
\[
x^2 + y^2 = 4k^2
\]
