Find the number of common tangents to the circle `x^2 +y^2=4` and `x^2+y^2−6x−8y−24=0 `

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 can 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 identify:
- Center \( C_1 = (0, 0) \)
- Radius \( r_1 = \sqrt{4} = 2 \)

### Step 2: Rewrite the second circle in standard form
The second circle is given by the equation:
\[ x^2 + y^2 - 6x - 8y - 24 = 0 \]
We can rearrange this equation to find its center and radius. Completing the square for \( x \) and \( y \):

1. For \( x \):
\[ x^2 - 6x = (x - 3)^2 - 9 \]
2. For \( y \):
\[ y^2 - 8y = (y - 4)^2 - 16 \]

Substituting these back into the equation:
\[ (x - 3)^2 - 9 + (y - 4)^2 - 16 - 24 = 0 \]
This simplifies to:
\[ (x - 3)^2 + (y - 4)^2 - 49 = 0 \]
Thus, we have:
\[ (x - 3)^2 + (y - 4)^2 = 49 \]
From this, we can identify:
- Center \( C_2 = (3, 4) \)
- Radius \( r_2 = \sqrt{49} = 7 \)

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

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

Now we check:
- \( r_2 - r_1 = 7 - 2 = 5 \)

### Step 5: Analyze the conditions for common tangents
We have:
- \( d = 5 \)
- \( r_2 - r_1 = 5 \)

Since \( d = r_2 - r_1 \), the circles are internally tangent to each other. When two circles are internally tangent, they have exactly one common tangent.

### Conclusion
Thus, the number of common tangents to the two circles is:
\[ \text{Number of common tangents} = 1 \]