Converting this into the standard form $$(x-\frac 12)^2+(y-\frac 12)^2=\frac 14 \left(\frac{3x+4y-1}{5}\right)^2$$
$$\sqrt {\left (x-\frac 12 \right )^2+\left (y-\frac 12 \right )^2}=\frac 12 \left (\frac{3x+4y-1}{5} \right )$$
So the focus is $(\frac 12, \frac 12)$ and eccentricity is $\frac 12$
The distance of focus from directrix is $a(\frac 1e-e)$
So $$a(2-\frac 12) =\frac {1.5+2-1}{5}$$ $$a\frac 32 =\frac12$$ $$a=\frac 13$$
And $$b^2=\frac 19 (\frac 34)$$ $$b^2=\frac {1}{12}$$
Then $$LR=\frac {2b^2}{a}=\frac{1}{18}$$
I think I am making a coneptual mistake in the equation of ellipse, but I can’t pinpoint it.
Lastly $$LR=\frac{2b^2}{a}=\frac{2/12}{1/3}=\frac{1}{2}.$$