The actual question itself asks for the ratio between the smaller circle to the larger circle and the answer is C. $3-2\sqrt{2}$.
What I do not understand is how we are meant to get the radius of the smaller circle in terms of the radius of the larger circle as from there it would be comparatively simple to calculate the ratio between the two.
So far I have tried to treat the tangent points with the diameters as if they were in the center of the radius of the larger circle and that the intersection of two perpendicular lines coming off of this point would be the center of the circle. But this got me $\frac{1}{16}$ and nowhere close to any of the answers. Other than that I used the process of elimination to determine that of the choices provided C and E were the only ones possible.
The question along with a diagram is shown below.
Best Answer
HINTS:
Drawing a figure is a necessary first step.
$$ d( d+2r)=r^2,~~ d+2r =R~$$
Eliminate $d$
$$ R( R-2r)= r^2 $$
By which theorem of circles is the above result obtained?
$$x=r/R ~\text {= ratio of radii}, ~~x^2+2x-1=0 $$
$$ x=\pm\sqrt{2} -1 $$
Which root to choose? Why?
How do you get area ratio from ratio of radii $x$?
Also a more straight forward approach is suggested by @John Omielan using Pythagoras theorem, please see his comment.