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

and length of minor axis, $b=\frac{5}{\sqrt{3}}$