For the two circles `x^2+y^2=16 and x^2+y^2-2y=0`, there is/are

Options

A. one pair of common tangents
B. two pairs of common tangents
C. three common tangents
D. no common tangents

Explanation

To solve the problem of determining the relationship between the two 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 the standard form:
\[ x^2 + y^2 - 16 = 0 \]

The second circle is given by:
\[ x^2 + y^2 - 2y = 0 \]
This can be rewritten as:
\[ x^2 + (y^2 - 2y) = 0 \]
Completing the square for the \(y\) terms:
\[ x^2 + (y - 1)^2 - 1 = 0 \implies x^2 + (y - 1)^2 = 1 \]

### Step 2: Determine the centers and radii of the circles
For the first circle \(C_1: x^2 + y^2 - 16 = 0\):
- Center \(C_1 = (0, 0)\)
- Radius \(r_1 = \sqrt{16} = 4\)

For the second circle \(C_2: x^2 + (y - 1)^2 = 1\):
- Center \(C_2 = (0, 1)\)
- Radius \(r_2 = \sqrt{1} = 1\)

### Step 3: Calculate the distance between the centers of the circles
The distance \(d\) between the centers \(C_1\) and \(C_2\) can be calculated using the distance formula:
\[ d = \sqrt{(0 - 0)^2 + (1 - 0)^2} = \sqrt{1} = 1 \]

### Step 4: Analyze the relationship between the circles
To determine the relationship between the two circles, we need to compare the distance \(d\) with the sum and difference of the radii:
- Sum of the radii: \(r_1 + r_2 = 4 + 1 = 5\)
- Difference of the radii: \(r_1 - r_2 = 4 - 1 = 3\)

### Step 5: Apply the conditions for circle intersection
We have:
- \(d = 1\)
- \(r_1 + r_2 = 5\)
- \(r_1 - r_2 = 3\)

Now we check the conditions:
1. If \(d > r_1 + r_2\): Circles are separate.
2. If \(d = r_1 + r_2\): Circles are externally tangent.
3. If \(d < r_1 + r_2\) and \(d > |r_1 - r_2|\): Circles intersect at two points.
4. If \(d = |r_1 - r_2|\): Circles are internally tangent.
5. If \(d < |r_1 - r_2|\): One circle is inside the other without touching.

In our case:
- \(d = 1 < 3 = |r_1 - r_2|\)

This means that the first circle (radius 4) completely encloses the second circle (radius 1) without touching it.

### Conclusion
Since one circle is completely inside the other without touching, there are no common tangents between the two circles.

**Final Answer:** There are no common tangents.

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