The number of common tangents to `x^(2)+y^(2)=4,(x-3)^(2)+(y-4)^(2)=9` is

Options

A. 1
B. 2
C. 3
D. 0

Answer

To find the number of common tangents to the circles given by the equations \(x^2 + y^2 = 4\) and \((x - 3)^2 + (y - 4)^2 = 9\), we will follow these steps:

### Step 1: Identify the centers and radii of the circles

1. The first circle is given by the equation \(x^2 + y^2 = 4\).
- Center \(C_1 = (0, 0)\)
- Radius \(r_1 = \sqrt{4} = 2\)

2. The second circle is given by the equation \((x - 3)^2 + (y - 4)^2 = 9\).
- Center \(C_2 = (3, 4)\)
- Radius \(r_2 = \sqrt{9} = 3\)

### Step 2: Calculate the distance between the centers of the circles

To 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 3: Determine the relationship between the distance and the radii

Now we compare the distance \(d\) with the sum of the radii \(r_1 + r_2\):

\[ r_1 + r_2 = 2 + 3 = 5 \]

### Step 4: Analyze the relationship

Since \(d = r_1 + r_2\), this indicates that the circles touch externally. When two circles touch externally, there are exactly 3 common tangents (2 external tangents and 1 internal tangent).

### Conclusion

Thus, the number of common tangents to the given circles is **3**.