The number of common tangents to the circles
`x^(2)+y^(2)+2x+8y-23=0` and
`x^(2)+y^(2)-4x-10y+9=0` are

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the number of common tangents to the circles given by the equations:

1. \( x^2 + y^2 + 2x + 8y - 23 = 0 \)
2. \( x^2 + y^2 - 4x - 10y + 9 = 0 \)

we will follow these steps:

### Step 1: Rewrite the equations in standard form
We start by rewriting both equations in the standard form of a circle, which is \((x - h)^2 + (y - k)^2 = r^2\).

**For the first circle:**
\[ x^2 + y^2 + 2x + 8y - 23 = 0 \]
Rearranging gives:
\[ x^2 + 2x + y^2 + 8y = 23 \]
Completing the square:
\[ (x^2 + 2x + 1) + (y^2 + 8y + 16) = 23 + 1 + 16 \]
\[ (x + 1)^2 + (y + 4)^2 = 40 \]
Thus, the center \(C_1\) is \((-1, -4)\) and the radius \(r_1 = \sqrt{40} = 2\sqrt{10}\).

**For the second circle:**
\[ x^2 + y^2 - 4x - 10y + 9 = 0 \]
Rearranging gives:
\[ x^2 - 4x + y^2 - 10y = -9 \]
Completing the square:
\[ (x^2 - 4x + 4) + (y^2 - 10y + 25) = -9 + 4 + 25 \]
\[ (x - 2)^2 + (y - 5)^2 = 20 \]
Thus, the center \(C_2\) is \((2, 5)\) and the radius \(r_2 = \sqrt{20} = 2\sqrt{5}\).

### Step 2: Calculate the distance between the centers
Now we calculate the distance \(d\) between the centers \(C_1(-1, -4)\) and \(C_2(2, 5)\):
\[ d = \sqrt{(2 - (-1))^2 + (5 - (-4))^2} \]
\[ d = \sqrt{(2 + 1)^2 + (5 + 4)^2} = \sqrt{3^2 + 9^2} = \sqrt{9 + 81} = \sqrt{90} = 3\sqrt{10} \]

### Step 3: Determine the relationship between the circles
We need to compare \(d\) with \(r_1 + r_2\) and \(|r_1 - r_2|\):
\[ r_1 + r_2 = 2\sqrt{10} + 2\sqrt{5} \]
\[ |r_1 - r_2| = |2\sqrt{10} - 2\sqrt{5|} = 2(\sqrt{10} - \sqrt{5}) \]

### Step 4: Analyze the conditions
1. If \(d > r_1 + r_2\), the circles do not intersect and there are 4 common tangents.
2. If \(d = r_1 + r_2\), the circles touch externally and there are 3 common tangents.
3. If \(|r_1 - r_2| < d < r_1 + r_2\), the circles intersect and there are 2 common tangents.
4. If \(d = |r_1 - r_2|\), the circles touch internally and there is 1 common tangent.
5. If \(d < |r_1 - r_2|\), one circle is inside the other and there are no common tangents.

### Step 5: Conclusion
Now we compare:
- \(d = 3\sqrt{10}\)
- \(r_1 + r_2 = 2\sqrt{10} + 2\sqrt{5}\)

Since \(3\sqrt{10} > 2\sqrt{10} + 2\sqrt{5}\), the circles do not intersect.

Thus, the number of common tangents is **4**.