To solve the problem, we need to find the reflection of the given ellipse in the mirror line and then compare it with the given equation to find the values of \( k_1 \) and \( k_2 \).
### Step-by-Step Solution:
1. **Identify the Ellipse Equation:**
The given ellipse is:
\[
\frac{(x-4)^2}{16} + \frac{(y-3)^2}{9} = 1
\]
This can be rewritten in standard form as:
\[
(x - 4)^2 + \frac{(y - 3)^2}{\frac{9}{16}} = 1
\]
Here, \( h = 4 \), \( k = 3 \), \( a = 4 \), and \( b = 3 \).
2. **Finding a Point on the Ellipse:**
A point on the ellipse can be expressed as:
\[
(x, y) = (4 + 4 \cos \theta, 3 + 3 \sin \theta)
\]
3. **Equation of the Mirror Line:**
The mirror line is given by:
\[
x - y - 2 = 0
\]
This can be rewritten as:
\[
x - y = 2
\]
4. **Finding the Image of the Point:**
To find the reflection of the point \( (4 + 4 \cos \theta, 3 + 3 \sin \theta) \) across the line \( x - y - 2 = 0 \), we use the formula for reflection across a line \( Ax + By + C = 0 \):
\[
h' = \frac{(b^2 - a^2)x_1 - 2ab y_1 - 2ac}{a^2 + b^2}
\]
\[
k' = \frac{(a^2 - b^2)y_1 - 2ab x_1 - 2bc}{a^2 + b^2}
\]
Here, \( A = 1, B = -1, C = -2 \), \( x_1 = 4 + 4 \cos \theta \), and \( y_1 = 3 + 3 \sin \theta \).
5. **Calculating the Reflection:**
After applying the reflection formulas, we get:
\[
h' = 3 + 5 \sin \theta
\]
\[
k' = 4 \cos \theta + 2
\]
6. **Finding the Locus of the Reflected Points:**
We now express the locus of the reflected points \( (h', k') \) in terms of \( x \) and \( y \):
\[
x - 5 = 3 \sin \theta \quad \text{and} \quad y - 2 = 4 \cos \theta
\]
Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we substitute:
\[
\left(\frac{x - 5}{3}\right)^2 + \left(\frac{y - 2}{4}\right)^2 = 1
\]
7. **Expanding the Locus Equation:**
Expanding this equation gives:
\[
16(x - 5)^2 + 9(y - 2)^2 = 144
\]
Simplifying leads to:
\[
16x^2 + 9y^2 - 160x - 36y + 292 = 0
\]
8. **Comparing with Given Equation:**
The reflected ellipse is given in the form:
\[
k_1 x^2 + k_2 y^2 - 160x - 36y + 292 = 0
\]
Comparing coefficients, we find:
\[
k_1 = 16, \quad k_2 = 9
\]
9. **Calculating \( \frac{k_1 + k_2}{5} \):**
Now, we calculate:
\[
\frac{k_1 + k_2}{5} = \frac{16 + 9}{5} = \frac{25}{5} = 5
\]
### Final Answer:
Thus, the value of \( \frac{k_1 + k_2}{5} \) is \( 5 \).
