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

Explanation

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 centers and radii of the circles

1. **First Circle**: The equation is \(x^2 + y^2 = 4\).
- This can be rewritten as \((x - 0)^2 + (y - 0)^2 = 2^2\).
- Center: \(C_1(0, 0)\)
- Radius: \(r_1 = 2\)

2. **Second Circle**: The equation is \(x^2 + y^2 - 6x - 8y = 24\).
- Rearranging gives \(x^2 - 6x + y^2 - 8y = 24\).
- Completing the square:
- For \(x\): \(x^2 - 6x = (x - 3)^2 - 9\)
- For \(y\): \(y^2 - 8y = (y - 4)^2 - 16\)
- Thus, the equation becomes \((x - 3)^2 + (y - 4)^2 - 25 = 0\) or \((x - 3)^2 + (y - 4)^2 = 25\).
- Center: \(C_2(3, 4)\)
- Radius: \(r_2 = 5\)

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

- The distance \(d\) between the centers \(C_1(0, 0)\) and \(C_2(3, 4)\) is calculated using the distance formula:
\[ d = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 3: Determine the relationship between the distance and the radii

- The relationship between the distance \(d\), the radii \(r_1\) and \(r_2\) determines the number of common tangents:
- If \(d > r_1 + r_2\), there are 4 common tangents.
- If \(d = r_1 + r_2\), there are 3 common tangents (the circles touch externally).
- If \(|r_1 - r_2| < d < r_1 + r_2\), there are 2 common tangents.
- If \(d = |r_1 - r_2|\), there is 1 common tangent (the circles touch internally).
- If \(d < |r_1 - r_2|\), there are 0 common tangents (the circles are one inside the other without touching).

### Step 4: Evaluate the conditions

- Here, we have:
- \(r_1 = 2\)
- \(r_2 = 5\)
- \(d = 5\)

- Now, calculate \(r_1 + r_2\):
\[ r_1 + r_2 = 2 + 5 = 7 \]
- Since \(d = 5 < 7\), we check \(|r_1 - r_2|\):
\[ |r_1 - r_2| = |2 - 5| = 3 \]
- Since \(d = 5 > 3\), we conclude that \(d\) is between \(|r_1 - r_2|\) and \(r_1 + r_2\).

### Conclusion

Thus, there are **2 common tangents** to the circles.

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