The number of common tangents to `x^(2)+y^(2)=4, x^(2)+y^(2)-6x-8y-24=0` is

Options

A. one
B. two
C. four
D. zero

Answer

To find the number of common tangents to the circles given by the equations \( x^2 + y^2 = 4 \) and \( x^2 + y^2 - 6x - 8y - 24 = 0 \), we will follow these steps:

### Step 1: Identify the first circle
The first equation is:
\[ x^2 + y^2 = 4 \]
This represents a circle with:
- Center \( C_1 = (0, 0) \)
- Radius \( r_1 = \sqrt{4} = 2 \)

### Step 2: Identify the second circle
The second equation can be rewritten as:
\[ x^2 + y^2 - 6x - 8y - 24 = 0 \]
To find the center and radius, we complete the square for \( x \) and \( y \).

Rearranging the equation:
\[ (x^2 - 6x) + (y^2 - 8y) = 24 \]
Completing the square:
\[ (x - 3)^2 - 9 + (y - 4)^2 - 16 = 24 \]
This simplifies to:
\[ (x - 3)^2 + (y - 4)^2 = 49 \]
Thus, the second circle has:
- Center \( C_2 = (3, 4) \)
- Radius \( r_2 = \sqrt{49} = 7 \)

### Step 3: Calculate the distance between the centers
Now, we find the distance \( d \) between the centers \( C_1 \) and \( C_2 \):
\[ d = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 4: Analyze the relationship between the circles
To determine the number of common tangents, we compare the distance \( d \) with the radii \( r_1 \) and \( r_2 \):
- \( r_1 = 2 \)
- \( r_2 = 7 \)

We check the condition:
\[ d = r_2 - r_1 \]
Calculating:
\[ 7 - 2 = 5 \]
Since \( d = 5 \), which is equal to \( r_2 - r_1 \), this indicates that the circles touch internally.

### Conclusion
When two circles touch internally, there is exactly one common tangent.

Thus, the number of common tangents to the given circles is:
\[ \boxed{1} \]