What´s the value of the ratio $x$ in terms of the ratio $R$ in the figure below?

Question

For refrence: If $P$, $Q$ and $T$ are points of tangency, then $x$ in terms of $R$ is

My progress:

$\triangle JOT: (R+x)^2=x^2+(R+LT)^2\implies\\ R^2+x^2+2Rx = x^2+R^2+LT^2+2RLT\\ x^2+2Rx = R^2+2RLT\\ LT = \frac{x^2+2Rx-R^2}{2RLT}$

....??

Answer 1

Apply Pythagoras in $\triangle JOH$ with $JO = R + x, JH = x$

Apply Pythagoras in $\triangle AOH$ with $~~~~~~~AO = 2R - x, AH = R + x$

$OH^2 = (2R-x)^2 - (R+x)^2 = (R+x)^2 - x^2$

$ \implies x = \dfrac R4$