If `x:y:z=1:2:3`, then what is the value of `((3x^2-2y^2+4z^2)/(x^2+2y^2+z^2))`?

Options

A. `31/21`
B. `31/16`
C. `31/18`
D. `31/19`

Answer

To solve the problem, we start with the given ratio \( x:y:z = 1:2:3 \). This means we can express \( x \), \( y \), and \( z \) in terms of a common variable \( k \):

1. **Assign values based on the ratio**:
\[ x = k, \quad y = 2k, \quad z = 3k \]

2. **Substitute these values into the expression**:
We need to evaluate:
\[ \frac{3x^2 - 2y^2 + 4z^2}{x^2 + 2y^2 + z^2} \]

**Calculate the numerator**:
\[ 3x^2 = 3(k^2) = 3k^2 \]
\[ 2y^2 = 2(2k)^2 = 2 \cdot 4k^2 = 8k^2 \]
\[ 4z^2 = 4(3k)^2 = 4 \cdot 9k^2 = 36k^2 \]
Now, substituting these into the numerator:
\[ 3x^2 - 2y^2 + 4z^2 = 3k^2 - 8k^2 + 36k^2 = (3 - 8 + 36)k^2 = 31k^2 \]

**Calculate the denominator**:
\[ x^2 = k^2 \]
\[ 2y^2 = 2(2k)^2 = 8k^2 \]
\[ z^2 = (3k)^2 = 9k^2 \]
Now, substituting these into the denominator:
\[ x^2 + 2y^2 + z^2 = k^2 + 8k^2 + 9k^2 = (1 + 8 + 9)k^2 = 18k^2 \]

3. **Combine the results**:
Now we can substitute the numerator and denominator back into the expression:
\[ \frac{31k^2}{18k^2} \]

4. **Simplify**:
The \( k^2 \) terms cancel out:
\[ \frac{31}{18} \]

Thus, the final answer is:
\[ \frac{31}{18} \]