A sphere $x^{2}+y^{2}+z^{2}=9$ is cut by the plane $x+y+z=3$. The radius of the circle so formed is:

Question

Accepted Answer

To determine the radius of the circle formed by the intersection of the sphere $x^2 + y^2 + z^2 = 9$ and the plane $x + y + z = 3$, we can use the following approach:

1. **Equation of the Sphere**: The given sphere is centered at the origin $(0, 0, 0)$ with a radius of 3, as the equation $x^2 + y^2 + z^2 = 9$ can be rewritten as $(x - 0)^2 + (y - 0)^2 + (z - 0)^2 = 3^2$.

2. **Equation of the Plane**: The plane is given by $x + y + z = 3$.

3. **Distance from the Center of the Sphere to the Plane**: The distance $d$ from the center of the sphere $(0, 0, 0)$ to the plane $x + y + z = 3$ can be calculated using the formula for the distance from a point to a plane:
   $$
   d = \frac{|ax_1 + by_1 + cz_1 + d|}{\sqrt{a^2 + b^2 + c^2}}
   $$
   where $a = 1$, $b = 1$, $c = 1$, $d = -3$, and the point is $(x_1, y_1, z_1) = (0, 0, 0)$. Thus,
   $$
   d = \frac{|1 \cdot 0 + 1 \cdot 0 + 1 \cdot 0 - 3|}{\sqrt{1^2 + 1^2 + 1^2}} = \frac{|-3|}{\sqrt{3}} = \frac{3}{\sqrt{3}} = \sqrt{3}
   $$

4. **Radius of the Circle of Intersection**: The radius $r$ of the circle formed by the intersection is given by the Pythagorean theorem in the context of the sphere:
   $$
   r = \sqrt{R^2 - d^2}
   $$
   where $R = 3$ is the radius of the sphere and $d = \sqrt{3}$ is the distance from the center of the sphere to the plane. Therefore,
   $$
   r = \sqrt{3^2 - (\sqrt{3})^2} = \sqrt{9 - 3} = \sqrt{6}
   $$

Thus, the radius of the circle formed by the intersection is $\sqrt{6}$, which corresponds to Option 2.

**Explanation of Incorrect Options**:

- **Option 1 (6)**: This option is incorrect because it suggests a radius larger than the sphere itself, which is not possible given the geometry of the problem.
  
- **Option 3 (3)**: This option is incorrect because it implies that the plane passes through the center of the sphere, which is not the case here. The plane is at a distance of $\sqrt{3}$ from the center.

- **Option 4 ($\sqrt{3}$)**: This option is incorrect because it represents the distance from the center of the sphere to the plane, not the radius of the circle of intersection.

Therefore, the correct answer is indeed Option 2, $\sqrt{6}$.