<s>[INST] Find the area of ​the shaded region in the figure below For reference:

In the figure $O$ is the center of the circle and its radius measures $a$ and $AQ = QB$. Calculate the area of ​​the shaded region.(Answer: $\frac{a^2}{4}(\pi-2)$)

correct figure

My progress:
If $AQ = BQ \implies \angle AQB=90^\circ$
Complete the square $AQBD$.
incorrect figure. incorrect figure, please do not consider it for any effect

$OC = r$ and $QC =R = AC.$
$O$ is centre of square.
$QO$ is angle bisector, therefore $\angle AQO$ is $45^\circ.$
$QD = R\sqrt2$
Considering $\triangle OCQ$,
$\displaystyle r^2+\left(\frac{R}{2}\right)^2=OQ^2\implies r^2+\frac{R^2}{4}=(R\sqrt2)^2$
$\therefore R = \dfrac{2r\sqrt7}{7}$
I don't see a solution...is it missing some information?
The book has another similar question but in this question $a = 2$ and answers match if we replace $a$ with $2$. Diagram below -

 [/INST] Here is a solution that I have so far. It does turn out after all the lengthy work that $\angle HOF \approx 90^\circ$. So there should be a way to show $\angle HOF = 90^\circ$ when $O$ lies on $AE$. But I have not been able to see a geometric solution yet.

If $R$ is the radius of the quarter circle and radius of the smaller circle is $a$,
$AS^2 = AG \cdot AE~$ i.e. $~(R-a)^2 = (AO + a) \cdot (AO-a)$ $\implies AO^2 = R^2 - 2aR + 2a^2$
Also, $~QF \cdot QE = QT^2~$ or $~(R-FE) \cdot R = a^2 $
$\implies FE = \dfrac{R^2-a^2}{R}, ~QF = \dfrac{a^2}{R}$
If $M$ is the midpoint of $FE$, $OM \perp FE$. Notice that $\triangle OEM \sim \triangle OAS~$. So,
$\displaystyle \frac{ME^2}{OE^2} = \frac{AS^2}{AO^2} \implies \frac{(R^2-a^2)^2}{4 a^2 R^2} = \frac{(R-a)^2}{R^2-2aR+2a^2}$
Simplifying, $2a^4 + 2a^3R + R^4 - 5a^2R^2 = 0$
$(R-a) (2a^3 + 4a^2R - aR^2 - R^3) = 0$
As $R = a$ is not a solution that we are interested in, we solve $2a^3 + 4a^2R - aR^2 - R^3 = 0$. WolframAlpha gives an approximate form solution of $R \approx 1.81361 a$. We then find $AO \approx 1.28917a, QF \approx 0.551386 a$
Next, $ \displaystyle AK = AS \cdot \frac{AE}{AO} = (1.81361 a - a) \cdot \frac {AO + a}{AO} \approx 1.44472a$
So, $QK = R - AK \approx 0.36889a$
$ \displaystyle \cos \angle AQE = \frac{QK}{QE} \approx 0.2034$
Applying law of cosine, $AF^2 = AQ^2 + QF^2 - 2 AQ \cdot QF \cdot \cos \angle AQE \approx 3.18641 a^2$
$AS^2 = AH \cdot AF = (AF - HF) \cdot AF$
We obtain $~HF \approx 1.41422 a, ~$ which is approximately $a \sqrt2$. With $HF \approx a \sqrt2$ and $OF = OH = a$, we have $\angle HOF \approx 90^\circ$.
That leads to the shaded area $ \displaystyle A \approx \frac{\pi}{4} \cdot a^2 - \frac 12 \cdot a^2 = \frac {a^2}{4} (\pi - 2)$
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