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Question Number 207543 by efronzo1 last updated on 18/May/24

Commented by efronzo1 last updated on 18/May/24

Answered by A5T last updated on 18/May/24

$$rH=\frac{R(2r+2\sqrt{r^2+H^2})}{2}=rR+R\sqrt{r^2+H^2}$$$$\Rightarrow r^2{(H-R)}^2=R^2(r^2+H^2)$$$$\Rightarrow r^2\left(H^2\right)+r^2{R}^2-2r^{2}HR=r^2{R}^2+R^2{H}^2$$$$\Rightarrow H^2(r^2-R^2)=2r^{2}HR$$$$\Rightarrow \frac{r^2-R^2}{r^2{R}^2}=\frac{2}{HR}\Rightarrow \frac{1}{R^2}-\frac{1}{r^2}=\frac{2}{HR}$$

Answered by mr W last updated on 18/May/24

$$\frac{R}{r}=\frac{\sqrt{{(H-R)}^2-R^2}}{H}=\frac{\sqrt{H(H-2R)}}{H}$$$$\frac{R^2}{r^2}=1-\frac{2R}{H}$$$$\Rightarrow \frac{1}{R^2}-\frac{1}{r^2}=\frac{2}{RH}$$

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