If l, m n denote the lengths of the intercepts made by the circle `x ^(2) +y ^(2) - 8x + 10y + 16 =0 ` on x-axis, y-axis and `y = -x` respectively, then ` l + m + n` is equal to `l + m + n (sqrt2 = 1.41)`

Answer

To solve the problem, we need to find the lengths of the intercepts \( l \), \( m \), and \( n \) made by the given circle on the x-axis, y-axis, and the line \( y = -x \) respectively. The equation of the circle is:

\[ x^2 + y^2 - 8x + 10y + 16 = 0 \]

### Step 1: Rewrite the Circle Equation
First, we rewrite the equation in standard form by completing the square.

1. Group the \( x \) and \( y \) terms:
\[ (x^2 - 8x) + (y^2 + 10y) + 16 = 0 \]

2. Complete the square for \( x \):
\[ x^2 - 8x = (x - 4)^2 - 16 \]

3. Complete the square for \( y \):
\[ y^2 + 10y = (y + 5)^2 - 25 \]

4. Substitute back into the equation:
\[ (x - 4)^2 - 16 + (y + 5)^2 - 25 + 16 = 0 \]
\[ (x - 4)^2 + (y + 5)^2 - 25 = 0 \]
\[ (x - 4)^2 + (y + 5)^2 = 25 \]

### Step 2: Identify the Center and Radius
From the standard form, we can identify the center and radius of the circle:
- Center \( (h, k) = (4, -5) \)
- Radius \( r = \sqrt{25} = 5 \)

### Step 3: Find the Length of the Intercept on the x-axis (\( l \))
The intercept on the x-axis occurs when \( y = 0 \):
1. Substitute \( y = 0 \) into the circle's equation:
\[ (x - 4)^2 + (-5)^2 = 25 \]
\[ (x - 4)^2 + 25 = 25 \]
\[ (x - 4)^2 = 0 \]
\[ x - 4 = 0 \implies x = 4 \]
The circle touches the x-axis at the point \( (4, 0) \), so the length of the intercept \( l = 0 \).

### Step 4: Find the Length of the Intercept on the y-axis (\( m \))
The intercept on the y-axis occurs when \( x = 0 \):
1. Substitute \( x = 0 \) into the circle's equation:
\[ (0 - 4)^2 + (y + 5)^2 = 25 \]
\[ 16 + (y + 5)^2 = 25 \]
\[ (y + 5)^2 = 9 \]
\[ y + 5 = \pm 3 \]
Thus, \( y = -2 \) or \( y = -8 \). The intercepts are at \( (0, -2) \) and \( (0, -8) \), so the length of the intercept \( m = | -2 - (-8) | = 6 \).

### Step 5: Find the Length of the Intercept on the line \( y = -x \) (\( n \))
To find the intercept on the line \( y = -x \), we need to find the perpendicular distance from the center of the circle to the line \( y = -x \):
1. The equation of the line can be rewritten as \( x + y = 0 \).
2. The formula for the distance \( d \) from a point \( (h, k) \) to the line \( Ax + By + C = 0 \) is:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
Here, \( A = 1, B = 1, C = 0 \), and the center \( (h, k) = (4, -5) \):
\[ d = \frac{|1 \cdot 4 + 1 \cdot (-5) + 0|}{\sqrt{1^2 + 1^2}} = \frac{|4 - 5|}{\sqrt{2}} = \frac{1}{\sqrt{2}} \]

3. The length of the intercept \( n \) can be found using the Pythagorean theorem:
\[ r^2 = \left(\frac{n}{2}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 \]
\[ 25 = \frac{n^2}{4} + \frac{1}{2} \]
\[ 25 - \frac{1}{2} = \frac{n^2}{4} \]
\[ \frac{49}{2} = \frac{n^2}{4} \]
\[ n^2 = 98 \implies n = 7\sqrt{2} \]

### Step 6: Calculate \( l + m + n \)
Now we can find \( l + m + n \):
\[ l + m + n = 0 + 6 + 7\sqrt{2} \]

### Final Answer
Thus, the final result is:
\[ l + m + n = 6 + 7\sqrt{2} \]