The radius of the circle in which the sphere $x^{2}+y^{2}+z^{2}=25$ is cut by the plane $2x+3y-6z=28$ will be :

Question

Accepted Answer

To determine the radius of the circle formed by the intersection of the sphere $x^2 + y^2 + z^2 = 25$ and the plane $2x + 3y - 6z = 28$, we need to follow these steps:

1. **Identify the Sphere and Plane:**
   - The given sphere is centered at the origin $(0, 0, 0)$ with a radius of 5, since $x^2 + y^2 + z^2 = 25$ implies $r^2 = 25$, so $r = 5$.

2. **Find the Distance from the Center of the Sphere to the Plane:**
   - The formula for the distance $d$ from a point $(x_0, y_0, z_0)$ to a plane $Ax + By + Cz + D = 0$ is:
     $$
     d = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}}
     $$
   - For the plane $2x + 3y - 6z = 28$, rewrite it as $2x + 3y - 6z - 28 = 0$.
   - The center of the sphere is $(0, 0, 0)$, so:
     $$
     d = \frac{|2(0) + 3(0) - 6(0) - 28|}{\sqrt{2^2 + 3^2 + (-6)^2}} = \frac{|-28|}{\sqrt{4 + 9 + 36}} = \frac{28}{\sqrt{49}} = \frac{28}{7} = 4
     $$

3. **Determine the Radius of the Circle:**
   - The radius $r_c$ of the circle formed by the intersection is given by the Pythagorean theorem in the context of the sphere:
     $$
     r_c = \sqrt{r^2 - d^2}
     $$
   - Here, $r = 5$ and $d = 4$, so:
     $$
     r_c = \sqrt{5^2 - 4^2} = \sqrt{25 - 16} = \sqrt{9} = 3
     $$

Thus, the radius of the circle is 3, which corresponds to Option 1.

**Explanation for Incorrect Options:**

- **Option 2 (13):** This value is incorrect because it exceeds the radius of the sphere itself, which is 5. The radius of the circle formed by the intersection cannot be greater than the radius of the sphere.

- **Option 3 (4):** This value is incorrect because it represents the distance from the center of the sphere to the plane, not the radius of the circle formed by the intersection.

- **Option 4 (5):** This value is incorrect because it represents the radius of the sphere, not the radius of the circle formed by the intersection. The circle's radius is determined by the Pythagorean relationship between the sphere's radius and the distance from the center to the plane.

Therefore, the correct answer is Option 1, with a radius of 3 for the circle formed by the intersection.