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Question-52506




Question Number 52506 by ajfour last updated on 09/Jan/19
Commented by ajfour last updated on 09/Jan/19
Find radii of both circles in terms  of a and b.
$${Find}\:{radii}\:{of}\:{both}\:{circles}\:{in}\:{terms} \\ $$$${of}\:{a}\:{and}\:{b}. \\ $$
Answered by mr W last updated on 09/Jan/19
(R−a)^2 =R(R−b)  R^2 −2aR+a^2 =R^2 −bR  (2a−b)R=a^2   ⇒R=(a^2 /(2a−b))    2R−2r=b  ⇒r=R−(b/2)  ⇒r=(a^2 /(2a−b))−(b/2)
$$\left({R}−{a}\right)^{\mathrm{2}} ={R}\left({R}−{b}\right) \\ $$$${R}^{\mathrm{2}} −\mathrm{2}{aR}+{a}^{\mathrm{2}} ={R}^{\mathrm{2}} −{bR} \\ $$$$\left(\mathrm{2}{a}−{b}\right){R}={a}^{\mathrm{2}} \\ $$$$\Rightarrow{R}=\frac{{a}^{\mathrm{2}} }{\mathrm{2}{a}−{b}} \\ $$$$ \\ $$$$\mathrm{2}{R}−\mathrm{2}{r}={b} \\ $$$$\Rightarrow{r}={R}−\frac{{b}}{\mathrm{2}} \\ $$$$\Rightarrow{r}=\frac{{a}^{\mathrm{2}} }{\mathrm{2}{a}−{b}}−\frac{{b}}{\mathrm{2}} \\ $$
Commented by ajfour last updated on 09/Jan/19
Best way Sir, thanks a lot!
$${Best}\:{way}\:{Sir},\:{thanks}\:{a}\:{lot}! \\ $$

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