If x:y=5:6, then `(3x^2 - 2y^2) : (y^2 - x^2)` is

Options

A. `7:6`
B. `11:3`
C. `3:11`
D. `6:7`

Answer

To solve the problem, we need to find the ratio \( (3x^2 - 2y^2) : (y^2 - x^2) \) given that \( x:y = 5:6 \).

### Step-by-Step Solution:

1. **Express x in terms of y**:
Since \( x:y = 5:6 \), we can express \( x \) in terms of \( y \):
\[ x = \frac{5}{6}y \]

2. **Substitute x in the expression**:
We need to substitute \( x \) in the expression \( (3x^2 - 2y^2) : (y^2 - x^2) \):
\[ 3x^2 = 3\left(\frac{5}{6}y\right)^2 = 3 \cdot \frac{25}{36}y^2 = \frac{75}{36}y^2 \]
\[ x^2 = \left(\frac{5}{6}y\right)^2 = \frac{25}{36}y^2 \]

3. **Calculate \( y^2 - x^2 \)**:
Now, we calculate \( y^2 - x^2 \):
\[ y^2 - x^2 = y^2 - \frac{25}{36}y^2 = \left(1 - \frac{25}{36}\right)y^2 = \frac{36}{36}y^2 - \frac{25}{36}y^2 = \frac{11}{36}y^2 \]

4. **Substitute into the ratio**:
Now substitute these values back into the ratio:
\[ (3x^2 - 2y^2) = \left(\frac{75}{36}y^2 - 2y^2\right) = \left(\frac{75}{36}y^2 - \frac{72}{36}y^2\right) = \frac{3}{36}y^2 = \frac{1}{12}y^2 \]

5. **Form the final ratio**:
Now we can form the final ratio:
\[ \frac{(3x^2 - 2y^2)}{(y^2 - x^2)} = \frac{\frac{1}{12}y^2}{\frac{11}{36}y^2} = \frac{1}{12} \cdot \frac{36}{11} = \frac{3}{11} \]

6. **Final Answer**:
Therefore, the ratio \( (3x^2 - 2y^2) : (y^2 - x^2) \) is:
\[ \frac{3}{11} \]