Given that a circle is tangent to two perpendicular diameters of a larger circle and the larger circle itself. What is the radius of the small circle?

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.

• What did you try? Commented Mar 20, 2023 at 21:20
• @ArcticChar I 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 1/16 and nowhere close to any of the answers. Other than that I knew it would be less than 1 and only C is less than 1 which implies that it was probably the answer and the answer key corroborates that. Commented Mar 20, 2023 at 21:39
• @EliCompton FYI, using an Approach0 search, there's Let R be the region of the disc $x^2+y^2\leq1$ in the first quadrant. Then the area of the largest possible circle contained in R. Note that the ratio of areas doesn't depend on what the outer circle's radius is. Thus, having it be $1$, with its center being at $(0,0)$, your smaller circle is the largest possible in that quadrant. Thus, as the linked answer indicates, ... Commented Mar 20, 2023 at 22:30
• @EliCompton (cont.) the center of the smaller circle is on the line joining the circles' tangent point to the larger circle's center. From this, you can determine a relationship between the $2$ ratios, then square it to get the resulting answer. Commented Mar 20, 2023 at 22:32

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.

• I believe it's somewhat easier to use the Pythagorean theorem to get, using your diagram and values, that $r+d=\sqrt{2}r$, so $R = (r+d)+r=(1+\sqrt{2})r$. Thus, one just needs to do something like multiply both sides by $\sqrt{2}-1$ and then square the result, plus there's no quadratic equation involved and, thus, need to solve it and choose a particular root. Commented Mar 20, 2023 at 22:41
• Yes indeed more straight forward! Commented Mar 20, 2023 at 22:47