Question Number 79794 by mr W last updated on 28/Jan/20
Commented by mr W last updated on 28/Jan/20
$${Find}\:{the}\:{radii}\:{of}\:{two}\:{circles}\:\left({if}\:{exist}\right) \\ $$$${which}\:{touch}\:{each}\:{other}\:{and}\:{touch}\:{the} \\ $$$${parabola}\:{and}\:{the}\:{y}−{axis}\:{respectively}. \\ $$
Commented by key of knowledge last updated on 28/Jan/20
$$\mathrm{C}\left\{\mathrm{O}\left(\mathrm{x}=\frac{\mathrm{a}}{\mathrm{1}+\sqrt{\frac{\mathrm{4a}^{\mathrm{2}} }{\mathrm{1}+\mathrm{4a}^{\mathrm{2}} }}}\:,\:\mathrm{y}=\left(\frac{\mathrm{a}−\mathrm{x}}{\mathrm{2a}}\right)+\mathrm{a}^{\mathrm{2}} \right)\:;\:\mathrm{r}=\mathrm{x}\:\right\} \\ $$$$\mathrm{circle}\:\mathrm{C}\:\mathrm{for}\:\forall\mathrm{a}\:\mathrm{touch}\:\mathrm{y}=\mathrm{x}^{\mathrm{2}} \: \\ $$$$\mathrm{and}\:\mathrm{if}\:\mathrm{circle}\:\mathrm{C}_{\mathrm{1}} \:\mathrm{and}\:\mathrm{C}_{\mathrm{2}} \:\mathrm{touch}\:\mathrm{each}\:\mathrm{other}\Rightarrow\mathrm{2}\sqrt{\mathrm{r}_{\mathrm{2}} \mathrm{r}_{\mathrm{1}} }=\mathrm{y}_{\mathrm{o2}} −\mathrm{y}_{\mathrm{o1}} \\ $$
Answered by ajfour last updated on 28/Jan/20
