Consider the spheres `x^2+y^2+z^2-4y+3=0 and x^2+y^2+z^2+2x+4z-4=0`
Consider the following statements
I. The two spheres intersect each other
II. The radius of first sphere is less than that of second sphere.
Which one of the above statement(s) is/are correct

Options

A. A. Only I
B. B. Only II
C. C. Both I and II
D. D. Neither I nor II

Answer

To solve the problem, we will analyze the equations of the two spheres and determine their centers and radii. We will then check whether the spheres intersect and compare their radii.

### Step 1: Rewrite the equations of the spheres

The equations of the spheres are given as:
1. \( x^2 + y^2 + z^2 - 4y + 3 = 0 \)
2. \( x^2 + y^2 + z^2 + 2x + 4z - 4 = 0 \)

### Step 2: Complete the square for the first sphere

For the first sphere:
\[ x^2 + (y^2 - 4y) + z^2 + 3 = 0 \]
To complete the square for \(y\):
\[ y^2 - 4y = (y - 2)^2 - 4 \]
Substituting this back, we get:
\[ x^2 + (y - 2)^2 - 4 + z^2 + 3 = 0 \]
This simplifies to:
\[ x^2 + (y - 2)^2 + z^2 = 1 \]

### Step 3: Identify the center and radius of the first sphere

From the equation \(x^2 + (y - 2)^2 + z^2 = 1\), we can identify:
- Center \(C_1 = (0, 2, 0)\)
- Radius \(R_1 = \sqrt{1} = 1\)

### Step 4: Complete the square for the second sphere

For the second sphere:
\[ x^2 + (y^2) + (z^2 + 4z) + 2x - 4 = 0 \]
To complete the square for \(x\) and \(z\):
\[ x^2 + 2x = (x + 1)^2 - 1 \]
\[ z^2 + 4z = (z + 2)^2 - 4 \]
Substituting these back, we get:
\[ (x + 1)^2 - 1 + y^2 + (z + 2)^2 - 4 - 4 = 0 \]
This simplifies to:
\[ (x + 1)^2 + y^2 + (z + 2)^2 = 9 \]

### Step 5: Identify the center and radius of the second sphere

From the equation \((x + 1)^2 + y^2 + (z + 2)^2 = 9\), we can identify:
- Center \(C_2 = (-1, 0, -2)\)
- Radius \(R_2 = \sqrt{9} = 3\)

### Step 6: Check if the spheres intersect

To determine if the spheres intersect, we need to check the distance between their centers and compare it to the sum of their radii.

1. **Distance between centers \(C_1\) and \(C_2\)**:
\[ d = \sqrt{(0 - (-1))^2 + (2 - 0)^2 + (0 - (-2))^2} \]
\[ = \sqrt{(1)^2 + (2)^2 + (2)^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \]

2. **Sum of the radii**:
\[ R_1 + R_2 = 1 + 3 = 4 \]

Since \(d = 3 < 4\), the spheres intersect.

### Step 7: Compare the radii

We have:
- \(R_1 = 1\)
- \(R_2 = 3\)

Clearly, \(R_1 < R_2\).

### Conclusion

Both statements are correct:
1. The two spheres intersect each other.
2. The radius of the first sphere is less than that of the second sphere.

Thus, the correct answer is that both statements are true.

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