If ` ( 2 x - y )^(2) + ( 3 y - 2 z )^(2)= 0 ` then the ratio ` x : y : z` is

Options

A. `1 : 3 : 2 `
B. ` 1 : 2 : 3 `
C. ` 3 : 1 : 2 `
D. ` 3 : 2 : 1 `

Answer

To solve the equation \( (2x - y)^2 + (3y - 2z)^2 = 0 \), we need to analyze the components of the equation.

### Step-by-step Solution:

1. **Understanding the Equation**:
The equation \( (2x - y)^2 + (3y - 2z)^2 = 0 \) implies that both terms must equal zero because the square of a real number is always non-negative. Therefore, we can set each part to zero:
\[ (2x - y)^2 = 0 \quad \text{and} \quad (3y - 2z)^2 = 0 \]

2. **Solving the First Equation**:
From \( (2x - y)^2 = 0 \), we have:
\[ 2x - y = 0 \]
Rearranging gives:
\[ y = 2x \]

3. **Solving the Second Equation**:
From \( (3y - 2z)^2 = 0 \), we have:
\[ 3y - 2z = 0 \]
Rearranging gives:
\[ 2z = 3y \quad \Rightarrow \quad z = \frac{3}{2}y \]

4. **Expressing Everything in Terms of x**:
Now, substitute \( y = 2x \) into \( z = \frac{3}{2}y \):
\[ z = \frac{3}{2}(2x) = 3x \]

5. **Finding the Ratios**:
Now we have:
- \( x = x \)
- \( y = 2x \)
- \( z = 3x \)

Therefore, the ratio \( x : y : z \) can be expressed as:
\[ x : 2x : 3x \]
Dividing each term by \( x \) (assuming \( x \neq 0 \)):
\[ 1 : 2 : 3 \]

### Final Answer:
Thus, the ratio \( x : y : z \) is \( 1 : 2 : 3 \).