Explanation:

- We are given that \(x^2 : y^2 = 49 : 64\). This implies \(\frac{x^2}{y^2} = \frac{49}{64}\).

- Therefore, \(\frac{x}{y} = \pm\frac{7}{8}\), considering the square root.

- Since x and y have opposite signs, choose \(\frac{x}{y} = -\frac{7}{8}\).

- Hence, \(x = -7k\) and \(y = 8k\) for some \(k\).

- Now calculate \(5x - 6y\):

$$

5x - 6y = 5(-7k) - 6(8k) = -35k - 48k = -83k

$$

- Calculate \(6x - 7y\):

$$

6x - 7y = 6(-7k) - 7(8k) = -42k - 56k = -98k

$$

- So, \(\frac{5x - 6y}{6x - 7y} = \frac{-83k}{-98k} = \frac{83}{98}\).

- Hence, the ratio (5x - 6y) : (6x - 7y) is 83 : 98.

- Option 4: 83 : 98 is correct.