In `Delta ABC` of a :b:c= 7 : 8: 9 .Then cos A : cos B is equal to :-

Options

A. `11/63`
B. `22/63`
C. `2/9`
D. `14/11`

Answer

To solve the problem of finding the ratio of cos A to cos B in triangle ABC where the sides are in the ratio a:b:c = 7:8:9, we can follow these steps:

### Step 1: Assign values to angles based on the given ratio
Let the sides opposite to angles A, B, and C be a, b, and c respectively. We can assign:
- a = 7k
- b = 8k
- c = 9k

### Step 2: Use the cosine rule to find cos A
The cosine rule states:
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]
Substituting the values of a, b, and c:
\[ \cos A = \frac{(8k)^2 + (9k)^2 - (7k)^2}{2 \cdot (8k) \cdot (9k)} \]
Calculating the squares:
\[ \cos A = \frac{64k^2 + 81k^2 - 49k^2}{144k^2} \]
Simplifying:
\[ \cos A = \frac{96k^2}{144k^2} = \frac{96}{144} = \frac{2}{3} \]

### Step 3: Use the cosine rule to find cos B
Now, we apply the cosine rule again for cos B:
\[ \cos B = \frac{a^2 + c^2 - b^2}{2ac} \]
Substituting the values:
\[ \cos B = \frac{(7k)^2 + (9k)^2 - (8k)^2}{2 \cdot (7k) \cdot (9k)} \]
Calculating the squares:
\[ \cos B = \frac{49k^2 + 81k^2 - 64k^2}{126k^2} \]
Simplifying:
\[ \cos B = \frac{66k^2}{126k^2} = \frac{66}{126} = \frac{11}{21} \]

### Step 4: Find the ratio of cos A to cos B
Now we can find the ratio:
\[ \frac{\cos A}{\cos B} = \frac{\frac{2}{3}}{\frac{11}{21}} \]
To divide fractions, we multiply by the reciprocal:
\[ \frac{\cos A}{\cos B} = \frac{2}{3} \cdot \frac{21}{11} = \frac{42}{33} = \frac{14}{11} \]

### Final Answer
Thus, the ratio of cos A to cos B is:
\[ \cos A : \cos B = 14 : 11 \]