In a triangle `ABC, (a)/(b) = (2)/(3) and sec^(2) A = (8)/(5)`. Find the number of triangle satisfying these conditions

Explanation

To solve the problem, we will follow these steps systematically.

### Step 1: Understand the given ratios and relationships
We are given that in triangle \( ABC \):
\[ \frac{a}{b} = \frac{2}{3} \]
This implies that \( a = 2k \) and \( b = 3k \) for some positive constant \( k \).

### Step 2: Use the secant relationship
We are also given that:
\[ \sec^2 A = \frac{8}{5} \]
From this, we can find \( \cos^2 A \):
\[ \cos^2 A = \frac{1}{\sec^2 A} = \frac{5}{8} \]

### Step 3: Use the cosine rule
Using the cosine rule, we know that:
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]
Substituting \( a = 2k \) and \( b = 3k \) into the cosine rule gives:
\[ \cos A = \frac{(3k)^2 + c^2 - (2k)^2}{2(3k)(c)} = \frac{9k^2 + c^2 - 4k^2}{6kc} = \frac{5k^2 + c^2}{6kc} \]

### Step 4: Set up the equation
Since we have \( \cos^2 A = \frac{5}{8} \), we can square the cosine expression:
\[ \left(\frac{5k^2 + c^2}{6kc}\right)^2 = \frac{5}{8} \]

### Step 5: Cross-multiply and simplify
Cross-multiplying gives:
\[ (5k^2 + c^2)^2 = \frac{5}{8} \cdot (36k^2c^2) \]
This simplifies to:
\[ (5k^2 + c^2)^2 = \frac{180k^2c^2}{8} = \frac{45k^2c^2}{2} \]

### Step 6: Expand and rearrange
Expanding the left side:
\[ 25k^4 + 10k^2c^2 + c^4 = \frac{45k^2c^2}{2} \]
Rearranging gives:
\[ c^4 + 25k^4 + 10k^2c^2 - \frac{45k^2c^2}{2} = 0 \]
Multiplying through by 2 to eliminate the fraction:
\[ 2c^4 + 50k^4 + 20k^2c^2 - 45k^2c^2 = 0 \]
This simplifies to:
\[ 2c^4 + 50k^4 - 25k^2c^2 = 0 \]

### Step 7: Factor the equation
Rearranging gives:
\[ 2c^4 - 25k^2c^2 + 50k^4 = 0 \]
Let \( x = c^2 \). The equation becomes:
\[ 2x^2 - 25k^2x + 50k^4 = 0 \]

### Step 8: Use the quadratic formula
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{25k^2 \pm \sqrt{(-25k^2)^2 - 4 \cdot 2 \cdot 50k^4}}{2 \cdot 2} \]
Calculating the discriminant:
\[ 625k^4 - 400k^4 = 225k^4 \]
Thus:
\[ x = \frac{25k^2 \pm 15k^2}{4} \]
This gives two solutions:
1. \( x = \frac{40k^2}{4} = 10k^2 \)
2. \( x = \frac{10k^2}{4} = \frac{5k^2}{2} \)

### Step 9: Determine the number of triangles
Since we have two distinct positive values for \( c^2 \) (i.e., \( 10k^2 \) and \( \frac{5k^2}{2} \)), there are two possible triangles that satisfy the given conditions.

### Final Answer
The number of triangles satisfying the given conditions is **2**.

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