Let the equation of the circle is `x^(2) + y^(2) = 25` and the equation of the line `x + y = 8`. If the radius of the circle of minimum area and also touches `x + y = 8` and `x^(2) + y^(2) = 25` is `(4sqrt(2) - 5)/(lambda)`. Then the value of `lambda` is.

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the given equations
The equation of the circle is given by:
\[ x^2 + y^2 = 25 \]
This represents a circle with a center at (0, 0) and a radius of 5 units (since \( \sqrt{25} = 5 \)).

The equation of the line is:
\[ x + y = 8 \]
This can be rewritten in intercept form as:
\[ \frac{x}{8} + \frac{y}{8} = 1 \]
indicating that it intersects the x-axis at (8, 0) and the y-axis at (0, 8).

### Step 2: Find the distance from the center of the circle to the line
To find the radius of the smaller circle that touches both the given circle and the line, we first need to calculate the distance from the center of the circle (0, 0) to the line \( x + y - 8 = 0 \).

The formula for the distance \( d \) from a point \( (x_1, y_1) \) to a line \( Ax + By + C = 0 \) is:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]

For our line \( x + y - 8 = 0 \):
- \( A = 1 \)
- \( B = 1 \)
- \( C = -8 \)
- The point is \( (0, 0) \).

Substituting these values into the distance formula:
\[ d = \frac{|1(0) + 1(0) - 8|}{\sqrt{1^2 + 1^2}} = \frac{|-8|}{\sqrt{2}} = \frac{8}{\sqrt{2}} = 4\sqrt{2} \]

### Step 3: Calculate the radius of the smaller circle
The radius of the smaller circle that touches the line and the original circle is given by:
\[ \text{Radius} = \text{Distance from center to line} - \text{Radius of original circle} \]
\[ \text{Radius} = 4\sqrt{2} - 5 \]

### Step 4: Relate the radius to the given expression
According to the problem, the radius can also be expressed as:
\[ \text{Radius} = \frac{4\sqrt{2} - 5}{\lambda} \]

Setting the two expressions for the radius equal to each other:
\[ 4\sqrt{2} - 5 = \frac{4\sqrt{2} - 5}{\lambda} \]

### Step 5: Solve for \( \lambda \)
To find \( \lambda \), we can multiply both sides by \( \lambda \):
\[ \lambda(4\sqrt{2} - 5) = 4\sqrt{2} - 5 \]

Assuming \( 4\sqrt{2} - 5 \neq 0 \), we can divide both sides by \( 4\sqrt{2} - 5 \):
\[ \lambda = 1 \]

### Final Answer
Thus, the value of \( \lambda \) is:
\[ \lambda = 1 \]