The radius of the circle in which the sphere `x^2+y^2+z^2+2x-2y-4z-19=0` is cut by the plane `x+2y+2z+7=0` is

Options

A. `2`
B. `3`
C. `4`
D. `1`

Answer

To find the radius of the circle formed by the intersection of the sphere and the plane, we can follow these steps:

### Step 1: Rewrite the Sphere Equation
The equation of the sphere is given as:
\[ x^2 + y^2 + z^2 + 2x - 2y - 4z - 19 = 0 \]
We can rearrange this equation to find the center and radius of the sphere. We will complete the square for \(x\), \(y\), and \(z\).

### Step 2: Completing the Square
1. For \(x^2 + 2x\):
\[ x^2 + 2x = (x + 1)^2 - 1 \]
2. For \(y^2 - 2y\):
\[ y^2 - 2y = (y - 1)^2 - 1 \]
3. For \(z^2 - 4z\):
\[ z^2 - 4z = (z - 2)^2 - 4 \]

Substituting these back into the sphere equation:
\[ (x + 1)^2 - 1 + (y - 1)^2 - 1 + (z - 2)^2 - 4 - 19 = 0 \]
This simplifies to:
\[ (x + 1)^2 + (y - 1)^2 + (z - 2)^2 - 25 = 0 \]
Thus, we have:
\[ (x + 1)^2 + (y - 1)^2 + (z - 2)^2 = 25 \]
From this, we can see that the center of the sphere is \((-1, 1, 2)\) and the radius \(r\) is \(5\).

### Step 3: Find the Plane Equation
The equation of the plane is given as:
\[ x + 2y + 2z + 7 = 0 \]

### Step 4: Calculate the Distance from the Center of the Sphere to the Plane
The distance \(d\) from a point \((x_0, y_0, z_0)\) to the plane \(Ax + By + Cz + D = 0\) is given by:
\[ d = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}} \]
For our plane:
- \(A = 1\), \(B = 2\), \(C = 2\), \(D = 7\)
- Center of the sphere \((-1, 1, 2)\)

Substituting these values into the distance formula:
\[ d = \frac{|1(-1) + 2(1) + 2(2) + 7|}{\sqrt{1^2 + 2^2 + 2^2}} \]
Calculating the numerator:
\[ = |-1 + 2 + 4 + 7| = |12| = 12 \]
Calculating the denominator:
\[ \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \]
Thus, the distance \(d\) is:
\[ d = \frac{12}{3} = 4 \]

### Step 5: Use Pythagorean Theorem to Find the Radius of the Circle
Let \(R\) be the radius of the circle formed by the intersection. By the Pythagorean theorem:
\[ R^2 + d^2 = r^2 \]
Where \(r\) is the radius of the sphere. Substituting the known values:
\[ R^2 + 4^2 = 5^2 \]
This simplifies to:
\[ R^2 + 16 = 25 \]
Thus:
\[ R^2 = 25 - 16 = 9 \]
Taking the square root gives:
\[ R = 3 \]

### Final Answer
The radius of the circle in which the sphere is cut by the plane is:
\[ \boxed{3} \]