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Question Number 19736 by ajfour last updated on 15/Aug/17

Commented by ajfour last updated on 15/Aug/17

$$\mathrm{Q}.19699\left(\mathrm{solution}\right)$$$$\mathrm{Find}\mathrm{r}\mathrm{in}\mathrm{terms}\mathrm{of}\mathrm{R}(=100)$$$$\mathrm{\angle }\mathrm{OXY}=30°.$$

Answered by ajfour last updated on 15/Aug/17

$$\mathrm{Let}\mathrm{radius}\mathrm{of}\mathrm{circle}{\mathrm{S}}_2\mathrm{with}\mathrm{centre}\mathrm{D}$$$$\mathrm{be}\mathrm{R}=100\left(\mathrm{given}\right)$$$$\mathrm{Radius}\mathrm{of}\mathrm{biggest}\mathrm{circle}\mathrm{S}\mathrm{is}2\mathrm{R}$$$$\mathrm{as}\mathrm{it}\mathrm{passes}\mathrm{through}\mathrm{O}\mathrm{and}\mathrm{touch}$$$$\mathrm{the}\mathrm{circle}\mathrm{S}\mathrm{internally}.$$$$\frac{\mathrm{DY}}{\mathrm{XD}}=\sin 30°\Rightarrow \frac{\mathrm{R}}{\mathrm{XD}}=\frac{1}{2}$$$$\mathrm{XD}=2\mathrm{R},\mathrm{hence}\mathrm{OX}=\mathrm{R}$$$$\mathrm{OC}=2\mathrm{R}-\mathrm{r}....\left(\mathrm{i}\right)$$$$\mathrm{CP}=\mathrm{MX}=\mathrm{OXcos}30°$$$$\Rightarrow \mathrm{CP}=\frac{\mathrm{R}\sqrt{3}}{2}....\left(\mathrm{ii}\right)$$$$\mathrm{OP}=\mathrm{OM}+\mathrm{MP}=\mathrm{OXsin}30°+\mathrm{CX}$$$$\Rightarrow \mathrm{OP}=\frac{\mathrm{R}}{2}+\mathrm{r}....\left(\mathrm{iii}\right)$$$$\mathrm{As}{\mathrm{OC}}^2={\mathrm{CP}}^2+{\mathrm{OP}}^2,\mathrm{using}\left(\mathrm{i}\right),$$$$\left(\mathrm{ii}\right),\mathrm{and}\left(\mathrm{iii}\right)\mathrm{we}\mathrm{can}\mathrm{write}$$$${(2\mathrm{R}-\mathrm{r})}^2=\frac{{3\mathrm{R}}^2}{4}+{(\frac{\mathrm{R}}{2}+\mathrm{r})}^2$$$$\Rightarrow {(2\mathrm{R}-\mathrm{r})}^2-{(\frac{\mathrm{R}}{2}+\mathrm{r})}^2=\frac{{3\mathrm{R}}^2}{4}$$$$\mathrm{or}\left(\frac{5\mathrm{R}}{2}\right)(\frac{3\mathrm{R}}{2}-2\mathrm{r})=\frac{{3\mathrm{R}}^2}{4}$$$$5(3\mathrm{R}-4\mathrm{r})=3\mathrm{R}$$$$15\mathrm{R}-20\mathrm{r}=3\mathrm{R}$$$$\mathrm{r}=\frac{12\mathrm{R}}{20}=\frac{3\mathrm{R}}{5}$$$$\mathrm{As}\mathrm{R}=100\mathrm{we}\mathrm{get}\mathrm{r}=60.$$

Commented by Tinkutara last updated on 15/Aug/17

$$\mathrm{Thank}\mathrm{you}\mathrm{very}\mathrm{much}\mathrm{Sir}!\mathrm{Thanks}\mathrm{for}$$$$\mathrm{your}\mathrm{efforts}!$$

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