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Question Number 4077 by Rasheed Soomro last updated on 28/Dec/15
For two co-planer circles to be tangent necessary and sufficient condition is, I think the distance between the centers of circles must be equal to r_1 +r_2 or ∣r_1 −r_2 ∣ , where r_1 and r_2 are the radii of the circles. If circles belong to different planes, what is necessary and sufficient condition for being tangent?
$$\mathrm{For}\:\mathrm{two}\:\boldsymbol{\mathrm{co}}-\boldsymbol{\mathrm{planer}}\:\mathrm{circles}\:\mathrm{to}\:\mathrm{be}\:\mathrm{tangent} \\ $$$$\:\mathrm{necessary}\:\mathrm{and}\:\mathrm{sufficient}\:\mathrm{condition}\:\mathrm{is}, \\ $$$$\mathrm{I}\:\mathrm{think} \\ $$$$\:\:\:\:\:\mathrm{the}\:\mathrm{distance}\:\mathrm{between}\:\mathrm{the}\:\mathrm{centers}\:\mathrm{of}\: \\ $$$$\:\:\:\:\mathrm{circles}\:\:\mathrm{must}\:\mathrm{be}\:\mathrm{equal}\:\mathrm{to}\:\boldsymbol{\mathrm{r}}_{\mathrm{1}} +\boldsymbol{\mathrm{r}}_{\mathrm{2}} \:\mathrm{or}\:\mid\boldsymbol{\mathrm{r}}_{\mathrm{1}} −\boldsymbol{\mathrm{r}}_{\mathrm{2}} \mid\:, \\ $$$$\:\:\:\:\:\mathrm{where}\:\:\boldsymbol{\mathrm{r}}_{\mathrm{1}} \mathrm{and}\:\boldsymbol{\mathrm{r}}_{\mathrm{2}} \:\mathrm{are}\:\mathrm{the}\:\mathrm{radii}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circles}. \\ $$$$\mathrm{If}\:\mathrm{circles}\:\mathrm{belong}\:\mathrm{to}\:\:\mathrm{different}\:\mathrm{planes},\:\mathrm{what} \\ $$$$\mathrm{is}\:\mathrm{necessary}\:\mathrm{and}\:\mathrm{sufficient}\:\mathrm{condition}\:\mathrm{for} \\ $$$$\mathrm{being}\:\mathrm{tangent}? \\ $$
Commented by Yozzii last updated on 27/Dec/15
Commented by Yozzii last updated on 27/Dec/15
Circles can touch internally as above. The broken lines are radii. ⇒r_1 +r_2 ≠C_1 C_2
$${Circles}\:{can}\:{touch}\:{internally}\:{as}\:{above}. \\ $$$${The}\:{broken}\:{lines}\:{are}\:{radii}.\: \\ $$$$\Rightarrow{r}_{\mathrm{1}} +{r}_{\mathrm{2}} \neq{C}_{\mathrm{1}} {C}_{\mathrm{2}} \\ $$$$ \\ $$
Commented by RasheedSindhi last updated on 28/Dec/15
You are right! I think I should add a word “externally”in my question then.
$$\mathrm{You}\:\mathrm{are}\:\mathrm{right}!\:\mathrm{I}\:\mathrm{think}\:\mathrm{I}\:\mathrm{should}\:\mathrm{add} \\ $$$$\mathrm{a}\:\mathrm{word}\:“\mathrm{externally}''\mathrm{in}\:\mathrm{my}\:\mathrm{question} \\ $$$$\mathrm{then}. \\ $$
Commented by Rasheed Soomro last updated on 28/Dec/15
If we don′t specify ′externally′ then the necessary and sufficient condition for being tangent of two circles should be,I think r_1 +r_2 =C_1 C_(2 ) ∣ ∣r_1 −r_2 ∣=C_1 C_(2 ) I have corrected the statement of the question.
$$\mathrm{If}\:\mathrm{we}\:\mathrm{don}'\mathrm{t}\:\mathrm{specify}\:'\mathrm{externally}'\:\mathrm{then} \\ $$$$\mathrm{the}\:\mathrm{necessary}\:\mathrm{and}\:\mathrm{sufficient}\:\mathrm{condition} \\ $$$$\mathrm{for}\:\mathrm{being}\:\mathrm{tangent}\:\mathrm{of}\:\mathrm{two}\:\mathrm{circles}\:\mathrm{should} \\ $$$$\mathrm{be},\mathrm{I}\:\mathrm{think} \\ $$$$\:\:\:\:\:\boldsymbol{\mathrm{r}}_{\mathrm{1}} +\boldsymbol{\mathrm{r}}_{\mathrm{2}} =\boldsymbol{\mathrm{C}}_{\mathrm{1}} \boldsymbol{\mathrm{C}}_{\mathrm{2}\:} \:\:\mid\:\:\:\mid\boldsymbol{\mathrm{r}}_{\mathrm{1}} −\boldsymbol{\mathrm{r}}_{\mathrm{2}} \mid=\boldsymbol{\mathrm{C}}_{\mathrm{1}} \boldsymbol{\mathrm{C}}_{\mathrm{2}\:} \\ $$$$\mathrm{I}\:\mathrm{have}\:\mathrm{corrected}\:\mathrm{the}\:\mathrm{statement}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{question}. \\ $$

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