The circles `x^(2)+y^(2)-10x +4y-20=0` and `x^(2)+y^(2)+14x-6y+22=0` touch internally.

Answer

To determine whether the circles touch internally, we will follow these steps:

### Step 1: Write the equations of the circles
The equations of the circles are given as:
1. Circle 1: \( x^2 + y^2 - 10x + 4y - 20 = 0 \)
2. Circle 2: \( x^2 + y^2 + 14x - 6y + 22 = 0 \)

### Step 2: Convert the equations to standard form
To convert the equations to standard form, we will complete the square for both circles.

**For Circle 1:**
\[ x^2 - 10x + y^2 + 4y - 20 = 0 \]
Completing the square:
- For \(x\): \(x^2 - 10x = (x - 5)^2 - 25\)
- For \(y\): \(y^2 + 4y = (y + 2)^2 - 4\)

Substituting back:
\[ (x - 5)^2 - 25 + (y + 2)^2 - 4 - 20 = 0 \]
\[ (x - 5)^2 + (y + 2)^2 - 49 = 0 \]
Thus, the standard form is:
\[ (x - 5)^2 + (y + 2)^2 = 49 \]
This gives us:
- Center \(C_1 = (5, -2)\)
- Radius \(r_1 = \sqrt{49} = 7\)

**For Circle 2:**
\[ x^2 + 14x + y^2 - 6y + 22 = 0 \]
Completing the square:
- For \(x\): \(x^2 + 14x = (x + 7)^2 - 49\)
- For \(y\): \(y^2 - 6y = (y - 3)^2 - 9\)

Substituting back:
\[ (x + 7)^2 - 49 + (y - 3)^2 - 9 + 22 = 0 \]
\[ (x + 7)^2 + (y - 3)^2 - 36 = 0 \]
Thus, the standard form is:
\[ (x + 7)^2 + (y - 3)^2 = 36 \]
This gives us:
- Center \(C_2 = (-7, 3)\)
- Radius \(r_2 = \sqrt{36} = 6\)

### Step 3: Calculate the distance between the centers
The distance \(d\) between the centers \(C_1\) and \(C_2\) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates:
\[ d = \sqrt{((-7) - 5)^2 + (3 - (-2))^2} \]
\[ d = \sqrt{(-12)^2 + (5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \]

### Step 4: Check the condition for internal tangency
For two circles to touch internally, the following condition must hold:
\[ d = r_1 - r_2 \]
Calculating \(r_1 - r_2\):
\[ r_1 - r_2 = 7 - 6 = 1 \]

### Step 5: Compare the distance and the difference of the radii
We found:
- Distance between centers \(d = 13\)
- Difference of radii \(r_1 - r_2 = 1\)

Since \(d \neq r_1 - r_2\), the circles do not touch internally.

### Conclusion
The circles do not touch internally.

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