The number of common tangents of the circles `x^(2) +y^(2) =16` and `x^(2) +y^(2) -2y = 0 ` is `:`

Options

A. 2
B. 3
C. 4
D. 0

Answer

To find the number of common tangents between the circles given by the equations \( x^2 + y^2 = 16 \) and \( x^2 + y^2 - 2y = 0 \), we will follow these steps:

### Step 1: Identify the equations of the circles
The first circle is given by:
\[ x^2 + y^2 = 16 \]
This can be rewritten in standard form as:
\[ (x - 0)^2 + (y - 0)^2 = 4^2 \]
Thus, the center \( C_1 \) is \( (0, 0) \) and the radius \( r_1 = 4 \).

The second circle is given by:
\[ x^2 + y^2 - 2y = 0 \]
We can rearrange this as:
\[ x^2 + (y^2 - 2y) = 0 \]
Completing the square for \( y \):
\[ x^2 + (y - 1)^2 - 1 = 0 \implies x^2 + (y - 1)^2 = 1^2 \]
Thus, the center \( C_2 \) is \( (0, 1) \) and the radius \( r_2 = 1 \).

### Step 2: Calculate the distance between the centers of the circles
The distance \( d \) between the centers \( C_1(0, 0) \) and \( C_2(0, 1) \) is calculated using the distance formula:
\[ d = \sqrt{(0 - 0)^2 + (1 - 0)^2} = \sqrt{0 + 1} = 1 \]

### Step 3: Compare the distance with the radii
Now we will compare the distance \( d \) with the radii \( r_1 \) and \( r_2 \):
- \( r_1 = 4 \)
- \( r_2 = 1 \)

We need to check the condition:
\[ d < |r_1 - r_2| \]
Calculating \( |r_1 - r_2| \):
\[ |r_1 - r_2| = |4 - 1| = 3 \]
Now we see that:
\[ 1 < 3 \]

### Step 4: Determine the position of the circles
Since \( d < |r_1 - r_2| \), this indicates that the second circle lies completely inside the first circle without touching it.

### Step 5: Conclusion about the common tangents
When one circle lies completely inside another without touching, there are no common tangents between the two circles.

Thus, the number of common tangents of the circles is:
\[ \boxed{0} \]