If `(x-3)^(2) + (y-5)^(2) + (z-4)^(2) =`0, then the value of `(x^2)/(9) + (y^2)/(25) + (z^2)/(16)` is

Options

A. 12
B. 9
C. 3
D. 1

Answer

To solve the given problem step by step, we need to analyze the equation provided and derive the required values.

### Step 1: Analyze the equation
The equation given is:
\[ (x-3)^{2} + (y-5)^{2} + (z-4)^{2} = 0 \]
Since the sum of squares is equal to zero, each individual square must also be zero. This leads us to the following equations:
\[ x - 3 = 0, \quad y - 5 = 0, \quad z - 4 = 0 \]

### Step 2: Solve for x, y, and z
From the equations derived:
1. \(x - 3 = 0 \Rightarrow x = 3\)
2. \(y - 5 = 0 \Rightarrow y = 5\)
3. \(z - 4 = 0 \Rightarrow z = 4\)

### Step 3: Substitute the values into the required expression
Now we need to find the value of:
\[ \frac{x^2}{9} + \frac{y^2}{25} + \frac{z^2}{16} \]
Substituting the values of \(x\), \(y\), and \(z\):
\[ \frac{3^2}{9} + \frac{5^2}{25} + \frac{4^2}{16} \]

### Step 4: Calculate each term
Now we calculate each term:
1. \(\frac{3^2}{9} = \frac{9}{9} = 1\)
2. \(\frac{5^2}{25} = \frac{25}{25} = 1\)
3. \(\frac{4^2}{16} = \frac{16}{16} = 1\)

### Step 5: Add the results
Adding these values together:
\[ 1 + 1 + 1 = 3 \]

### Final Answer
Thus, the value of \(\frac{x^2}{9} + \frac{y^2}{25} + \frac{z^2}{16}\) is:
\[ \boxed{3} \]