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Question Number 194270 by Tawa11 last updated on 01/Jul/23

Commented by mr W last updated on 02/Jul/23

$$R:r=3:2$$

Answered by mr W last updated on 02/Jul/23

Commented by mr W last updated on 02/Jul/23

$$OF=R-r$$$$FE=(R-r)\cos \theta$$$$OE=(R-r)\sin \theta$$$$CE=EB=\sqrt{r^2-{(R-r)}^2{\cos }^2\theta }$$$$b=CB=2\sqrt{r^2-{(R-r)}^2{\cos }^2\theta }$$$$a=AB=R-(R-r)\sin \theta -\sqrt{r^2-{(R-r)}^2{\cos }^2\theta }$$$$c=DC=R+(R-r)\sin \theta -\sqrt{r^2-{(R-r)}^2{\cos }^2\theta }$$$$a:b:c=2:7:3$$$$\Rightarrow a=2k,b=7k,c=3k$$$$R-(R-r)\sin \theta -\sqrt{r^2-{(R-r)}^2{\cos }^2\theta }=2k...\left(u\right)$$$$2\sqrt{r^2-{(R-r)}^2{\cos }^2\theta }=7k...\left(ii\right)$$$$R+(R-r)\sin \theta -\sqrt{r^2-{(R-r)}^2{\cos }^2\theta }=3k...\left(iii\right)$$$$$$$$\left(i\right):R-(R-r)\sin \theta =\frac{11k}{2}$$$$\left(iii\right):R+(R-r)\sin \theta =\frac{13k}{2}$$$$\Rightarrow R=6k$$$$\Rightarrow (R-r)\sin \theta =\frac{k}{2}$$$$$$$$2\sqrt{r^2-{(R-r)}^2{\cos }^2\theta }=7k$$$$2\sqrt{r^2-{(R-r)}^2+{(R-r)}^2{\sin }^2\theta }=7k$$$$\sqrt{2Rr-R^2+\frac{k^2}{4}}=\frac{7k}{2}$$$$\sqrt{12kr-36k^2+\frac{k^2}{4}}=\frac{7k}{2}$$$$\sqrt{48kr-143k^2}=7k$$$$\Rightarrow r=4k$$$$$$$$\Rightarrow \frac{R}{r}=\frac{6k}{4k}=\frac{3}{2}✓$$$$$$$$(R-r)\sin \theta =\frac{k}{2}$$$$(6k-4k)\sin \theta =\frac{k}{2}$$$$\sin \theta =\frac{1}{4}$$$$\Rightarrow \theta ={\sin }^{-1}\frac{1}{4}\approx 14.48°$$

Commented by Tawa11 last updated on 02/Jul/23

$$\mathrm{Wow},\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.\mathrm{I}\mathrm{really}\mathrm{appreciate}.$$

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