In triangle ABC given `9a ^(2) + 9b ^(2) - 17 c ^(2) =0.` If `(cot A + cot B)/(cot C) =m/n,` then the value of `(m+n)` equals

Answer

To solve the problem step by step, we will follow the reasoning presented in the video transcript.

### Step 1: Analyze the given equation
We start with the equation given in the problem:
\[ 9a^2 + 9b^2 - 17c^2 = 0 \]
This can be rearranged to:
\[ 9a^2 + 9b^2 = 17c^2 \]

### Step 2: Express \( a^2 + b^2 \)
From the rearranged equation, we can express \( a^2 + b^2 \) as:
\[ a^2 + b^2 = \frac{17}{9}c^2 \]

### Step 3: Use the cotangent formula
We need to find the value of \( \frac{\cot A + \cot B}{\cot C} \). Recall that:
\[ \cot A = \frac{\cos A}{\sin A}, \quad \cot B = \frac{\cos B}{\sin B}, \quad \cot C = \frac{\cos C}{\sin C} \]
Thus,
\[ \frac{\cot A + \cot B}{\cot C} = \frac{\frac{\cos A}{\sin A} + \frac{\cos B}{\sin B}}{\frac{\cos C}{\sin C}} = \frac{\cos A \sin C + \cos B \sin C}{\sin A \cos C} \]

### Step 4: Apply the Law of Sines
Using the Law of Sines:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R \]
This gives us:
\[ \sin A = \frac{a}{2R}, \quad \sin B = \frac{b}{2R}, \quad \sin C = \frac{c}{2R} \]

### Step 5: Substitute values for cosines
Using the cosine rule:
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc}, \quad \cos B = \frac{c^2 + a^2 - b^2}{2ca}, \quad \cos C = \frac{a^2 + b^2 - c^2}{2ab} \]

### Step 6: Substitute into the cotangent expression
Now substituting the values of \( \cos A, \cos B, \cos C, \sin A, \sin B, \sin C \) into the expression:
\[ \frac{\frac{b^2 + c^2 - a^2}{2bc} \cdot \frac{c}{2R} + \frac{c^2 + a^2 - b^2}{2ca} \cdot \frac{c}{2R}}{\frac{a^2 + b^2 - c^2}{2ab} \cdot \frac{c}{2R}} \]

### Step 7: Simplify the expression
After simplification, we will have:
\[ \frac{(b^2 + c^2 - a^2) + (c^2 + a^2 - b^2)}{a^2 + b^2 - c^2} \]
This simplifies to:
\[ \frac{2c^2}{a^2 + b^2 - c^2} \]

### Step 8: Substitute \( a^2 + b^2 \) from Step 2
Substituting \( a^2 + b^2 = \frac{17}{9}c^2 \):
\[ \frac{2c^2}{\frac{17}{9}c^2 - c^2} = \frac{2c^2}{\frac{17}{9}c^2 - \frac{9}{9}c^2} = \frac{2c^2}{\frac{8}{9}c^2} = \frac{2}{\frac{8}{9}} = \frac{2 \cdot 9}{8} = \frac{18}{8} = \frac{9}{4} \]

### Step 9: Identify \( m \) and \( n \)
From \( \frac{\cot A + \cot B}{\cot C} = \frac{9}{4} \), we have \( m = 9 \) and \( n = 4 \).

### Step 10: Calculate \( m + n \)
Thus, the value of \( m + n = 9 + 4 = 13 \).

### Final Answer
The final answer is:
\[ \boxed{13} \]