Find number of integral values of k for which the line `3x + 4y - k = 0` , lies between the circles `x^2 + y^2 - 2x - 2y + 1 = 0` and `x^2 + y^2 - 18 x - 12 y + 113 = 0`, without cutting a chord on either of circle.

Explanation

To find the number of integral values of \( k \) for which the line \( 3x + 4y - k = 0 \) lies between the circles defined by the equations \( x^2 + y^2 - 2x - 2y + 1 = 0 \) and \( x^2 + y^2 - 18x - 12y + 113 = 0 \) without cutting a chord on either circle, we can follow these steps:

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

1. **Circle 1**: \( x^2 + y^2 - 2x - 2y + 1 = 0 \)
- Rearranging gives: \( (x-1)^2 + (y-1)^2 = 1^2 \)
- Center \( C_1 = (1, 1) \), Radius \( R_1 = 1 \)

2. **Circle 2**: \( x^2 + y^2 - 18x - 12y + 113 = 0 \)
- Rearranging gives: \( (x-9)^2 + (y-6)^2 = 4^2 \)
- Center \( C_2 = (9, 6) \), Radius \( R_2 = 4 \)

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

The distance \( d \) between the centers \( C_1 \) and \( C_2 \) is given by:
\[ d = \sqrt{(9-1)^2 + (6-1)^2} = \sqrt{8^2 + 5^2} = \sqrt{64 + 25} = \sqrt{89} \]

### Step 3: Determine the conditions for the line to lie between the circles

For the line to lie between the circles without cutting a chord, the distance from the center of each circle to the line must satisfy the following inequalities:
\[ d_1 > R_1 \quad \text{and} \quad d_2 < d - R_2 \]

### Step 4: Calculate the distance from the center of each circle to the line

The distance \( d \) from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]

For our line \( 3x + 4y - k = 0 \):
- \( A = 3 \), \( B = 4 \), \( C = -k \)

1. **Distance from \( C_1(1, 1) \)**:
\[ d_1 = \frac{|3(1) + 4(1) - k|}{\sqrt{3^2 + 4^2}} = \frac{|7 - k|}{5} \]

2. **Distance from \( C_2(9, 6) \)**:
\[ d_2 = \frac{|3(9) + 4(6) - k|}{\sqrt{3^2 + 4^2}} = \frac{|27 + 24 - k|}{5} = \frac{|51 - k|}{5} \]

### Step 5: Set up the inequalities

1. For the first circle:
\[ \frac{|7 - k|}{5} > 1 \implies |7 - k| > 5 \]
This gives us two cases:
- \( 7 - k > 5 \implies k < 2 \)
- \( 7 - k < -5 \implies k > 12 \)

2. For the second circle:
\[ \frac{|51 - k|}{5} < \sqrt{89} - 4 \]
Calculate \( \sqrt{89} \approx 9.43 \):
\[ \sqrt{89} - 4 \approx 5.43 \implies |51 - k| < 27.15 \]
This gives us:
- \( 51 - k < 27.15 \implies k > 23.85 \)
- \( 51 - k > -27.15 \implies k < 78.15 \)

### Step 6: Combine the inequalities

From the inequalities:
1. \( k < 2 \) or \( k > 12 \)
2. \( k > 23.85 \) and \( k < 78.15 \)

The valid range for \( k \) is:
- \( k > 23.85 \) and \( k < 78.15 \)

### Step 7: Determine the integral values of \( k \)

The integral values of \( k \) satisfying \( 24 \leq k \leq 78 \) are:
\[ k = 24, 25, 26, \ldots, 78 \]

### Step 8: Count the integral values

The number of integral values is:
\[ 78 - 24 + 1 = 55 \]

### Final Answer
The number of integral values of \( k \) is \( 55 \).

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