If the radius of the circle touching the pair of lines `7x^(2) - 18 xy +7y^(2) = 0` and the circle `x^(2) +y^(2) - 8x - 8y = 0`, and contained in the given circle is equal to k, then `k^(2)` is equal to

Options

A. 10
B. 9
C. 8
D. 7

Explanation

To solve the problem step by step, we will follow the given information and derive the required value of \( k^2 \).

### Step 1: Identify the equations of the lines and the circle
The pair of lines is given by:
\[ 7x^2 - 18xy + 7y^2 = 0 \]
This can be factored to find the slopes of the lines.

The circle is given by:
\[ x^2 + y^2 - 8x - 8y = 0 \]

### Step 2: Rewrite the circle equation in standard form
We can rewrite the circle equation by completing the square:
\[ x^2 - 8x + y^2 - 8y = 0 \]
Completing the square for \( x \) and \( y \):
\[ (x - 4)^2 + (y - 4)^2 = 32 \]
This shows that the center of the circle is \( (4, 4) \) and the radius \( R \) is \( \sqrt{32} = 4\sqrt{2} \).

### Step 3: Find the angle bisectors of the lines
The lines represented by the equation \( 7x^2 - 18xy + 7y^2 = 0 \) can be analyzed. The angle bisectors of the lines can be derived from the coefficients. Since the coefficients of \( x^2 \) and \( y^2 \) are equal, the lines pass through the origin.

The angle bisectors can be given by the equations:
1. \( y = x \) (first bisector)
2. \( y = -x \) (second bisector)

### Step 4: Determine the center of the inscribed circle
Assuming the center of the inscribed circle lies on one of the angle bisectors, we can take \( (a, a) \) for the center of the circle. The equation of the circle can be written as:
\[ (x - a)^2 + (y - a)^2 = k^2 \]

### Step 5: Use the distance from the center to the line
The distance from the center \( (a, a) \) to one of the lines (for example, \( y = x \)) must equal the radius \( k \):
\[ \text{Distance} = \frac{|a - a|}{\sqrt{1 + (-1)^2}} = 0 \]
This indicates that the center is on the line.

### Step 6: Relate the radius \( k \) with the radius of the larger circle
The radius \( k \) of the inscribed circle must satisfy:
\[ R - k = \text{distance from center to the line} \]
Since the center is at \( (a, a) \) and lies on the angle bisector, we can derive:
\[ k = a - 0 \]

### Step 7: Calculate the radius using the relationship
We know \( R = 4\sqrt{2} \). The radius \( k \) of the inscribed circle can be derived from the relationship:
\[ 4\sqrt{2} - k = k \]
This leads to:
\[ 4\sqrt{2} = 2k \implies k = 2\sqrt{2} \]

### Step 8: Find \( k^2 \)
Now, we can find \( k^2 \):
\[ k^2 = (2\sqrt{2})^2 = 4 \times 2 = 8 \]

### Final Answer
Thus, the value of \( k^2 \) is:
\[ \boxed{8} \]