Step-by-step Solution:

The first two lines can be written as: \[ 2x - y + 2 = 0, \quad 2x - y - 2 = 0. \] The distance between them is: \[ a = \frac{2 - (-2)}{\sqrt{2^2 + 1^2}} = \frac{4}{\sqrt{5}}. \] The next two lines can be written as: \[ 4x - 3y - 5 = 0, \quad 4x - 3y + \frac{1}{2} = 0. \] The distance between them is: \[ b = \frac{\left| -5 - \frac{1}{2} \right|}{\sqrt{4^2 + 3^2}} = \frac{11}{10}. \] Hence, the ratio is: \[ \frac{a}{b} = \frac{\frac{4}{\sqrt{5}}}{\frac{11}{10}} = \frac{40}{11 \sqrt{5}}, \] or \[ 11 \sqrt{5} \, a = 40 \, b. \]