If `(tanx)/2=(tany)/3=(tanz)/5,x+y+z=pia n dtan^2x+tan^2y+tan^2z=(38)/Kt h a nK=_________`

Explanation

To solve the problem, we start with the given equations and relationships.

### Step 1: Set up the equations
We have:
\[ \frac{\tan x}{2} = \frac{\tan y}{3} = \frac{\tan z}{5} = t \]
From this, we can express \(\tan x\), \(\tan y\), and \(\tan z\) in terms of \(t\):
\[ \tan x = 2t, \quad \tan y = 3t, \quad \tan z = 5t \]

### Step 2: Use the angle sum property
Given that \(x + y + z = \pi\), we can use the tangent addition formula:
\[ \tan(x + y + z) = \tan(\pi) = 0 \]
Using the tangent addition formula for three angles, we have:
\[ \tan x + \tan y + \tan z = \tan x \tan y \tan z \]
Substituting the expressions for \(\tan x\), \(\tan y\), and \(\tan z\):
\[ 2t + 3t + 5t = (2t)(3t)(5t) \]
This simplifies to:
\[ 10t = 30t^3 \]

### Step 3: Rearranging the equation
Rearranging gives us:
\[ 30t^3 - 10t = 0 \]
Factoring out \(10t\):
\[ 10t(3t^2 - 1) = 0 \]
This gives us two possible cases:
1. \(t = 0\)
2. \(3t^2 - 1 = 0\)

### Step 4: Solve for \(t\)
From \(3t^2 - 1 = 0\):
\[ 3t^2 = 1 \implies t^2 = \frac{1}{3} \implies t = \frac{1}{\sqrt{3}} \text{ or } t = -\frac{1}{\sqrt{3}} \]
Since \(t\) represents the tangent of angles, we will consider \(t = \frac{1}{\sqrt{3}}\).

### Step 5: Calculate \(\tan^2 x + \tan^2 y + \tan^2 z\)
Now we can find \(\tan^2 x\), \(\tan^2 y\), and \(\tan^2 z\):
\[ \tan^2 x = (2t)^2 = 4t^2 = 4 \cdot \frac{1}{3} = \frac{4}{3} \]
\[ \tan^2 y = (3t)^2 = 9t^2 = 9 \cdot \frac{1}{3} = 3 \]
\[ \tan^2 z = (5t)^2 = 25t^2 = 25 \cdot \frac{1}{3} = \frac{25}{3} \]

Adding these together:
\[ \tan^2 x + \tan^2 y + \tan^2 z = \frac{4}{3} + 3 + \frac{25}{3} \]
Converting \(3\) into a fraction:
\[ 3 = \frac{9}{3} \]
Thus:
\[ \tan^2 x + \tan^2 y + \tan^2 z = \frac{4}{3} + \frac{9}{3} + \frac{25}{3} = \frac{38}{3} \]

### Step 6: Relate to the given equation
We know from the problem statement:
\[ \tan^2 x + \tan^2 y + \tan^2 z = \frac{38}{k} \]
Setting this equal to our result:
\[ \frac{38}{3} = \frac{38}{k} \]
This implies:
\[ k = 3 \]

### Final Answer
Thus, the value of \(k\) is:
\[ \boxed{3} \]