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

Options

A. 0
B. 1
C. 3
D. 4

Answer

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

### Step 1: Identify the first circle
The first circle is given by the equation:
\[ x^2 + y^2 = 4 \]
From this, we can determine that:
- Center \(C_1 = (0, 0)\)
- Radius \(R_1 = \sqrt{4} = 2\)

### Step 2: Identify the second circle
The second circle is given by the equation:
\[ x^2 + y^2 - 6x - 8y = 24 \]
We can rewrite this in standard form by completing the square:
\[ (x^2 - 6x) + (y^2 - 8y) = 24 \]
Completing the square:
\[ (x - 3)^2 - 9 + (y - 4)^2 - 16 = 24 \]
This simplifies to:
\[ (x - 3)^2 + (y - 4)^2 = 49 \]
From this, we find:
- Center \(C_2 = (3, 4)\)
- Radius \(R_2 = \sqrt{49} = 7\)

### Step 3: Calculate the distance between the centers
Now, we calculate 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: Compare the distance with the sum and difference of the radii
Next, we calculate the sum and difference of the radii:
- Sum of the radii: \(R_1 + R_2 = 2 + 7 = 9\)
- Difference of the radii: \(R_2 - R_1 = 7 - 2 = 5\)

### Step 5: Determine the relationship between the distance and the radii
Now we compare the distance \(d\) with the sum and difference of the radii:
- \(d = 5\)
- \(R_2 - R_1 = 5\)

Since the distance between the centers \(d\) is equal to the difference of the radii, this means that the circles touch each other internally.

### Conclusion: Number of common tangents
When two circles touch internally, they have exactly one common tangent.

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