The length of the minor axis of the ellipse (5x - 10) ^ 2 + (5y +...

{\left(x - 2\right)}^{2} + {\left(y + 3\right)}^{2} = {\left(\dfrac{1}{2}\dfrac{3x - 4y + 7}{5}\right)}^{2} is an ellipse, whose focus is $\left(2, - 3\right),$ directix $3x - 4y + 7 = 0$ and eccentricity is $\dfrac{1}{2}$
Length of $⊥$ from focus to directrix is $\dfrac{3 \times 2 - 4 \times \left(- 3\right) + 7}{5} = 5$
$\dfrac{a}{e} - ae = 5 ⇒ 2a - \dfrac{a}{2} = 5 ⇒ a = \dfrac{10}{3}$
So, length of major axis is $\dfrac{20}{3}$

and length of minor axis,~$b = \dfrac{5}{3}$