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Question Number 164941 by ajfour last updated on 23/Jan/22

Commented by ajfour last updated on 23/Jan/22

If radii of inscribed and circum-  scribed spheres are a and b, find  height and radius of cone.

$${If}\:{radii}\:{of}\:{inscribed}\:{and}\:{circum}- \\ $$$${scribed}\:{spheres}\:{are}\:\boldsymbol{{a}}\:{and}\:\boldsymbol{{b}},\:{find} \\ $$$${height}\:{and}\:{radius}\:{of}\:{cone}. \\ $$

Answered by mr W last updated on 23/Jan/22

height=h, radius=r  (2b−h)h=r^2    ...(i)  ((h−a)/a)=((√(h^2 +r^2 ))/( r))   ...(ii)  from (ii):  r^2 =((ha^2 )/((h−2a)))  ((ha^2 )/((h−2a)))=(2b−h)h  h^2 −2(a+b)h+a(a+4b)=0  ⇒h=a+b+(√(b(b−2a)))  r=a(√(h/(h−2a)))  ⇒r=a(√((b+a+(√(b(b−2a))))/(b−a+(√(b(b−2a))))))

$${height}={h},\:{radius}={r} \\ $$$$\left(\mathrm{2}{b}−{h}\right){h}={r}^{\mathrm{2}} \:\:\:...\left({i}\right) \\ $$$$\frac{{h}−{a}}{{a}}=\frac{\sqrt{{h}^{\mathrm{2}} +{r}^{\mathrm{2}} }}{\:{r}}\:\:\:...\left({ii}\right) \\ $$$${from}\:\left({ii}\right): \\ $$$${r}^{\mathrm{2}} =\frac{{ha}^{\mathrm{2}} }{\left({h}−\mathrm{2}{a}\right)} \\ $$$$\frac{{ha}^{\mathrm{2}} }{\left({h}−\mathrm{2}{a}\right)}=\left(\mathrm{2}{b}−{h}\right){h} \\ $$$${h}^{\mathrm{2}} −\mathrm{2}\left({a}+{b}\right){h}+{a}\left({a}+\mathrm{4}{b}\right)=\mathrm{0} \\ $$$$\Rightarrow{h}={a}+{b}+\sqrt{{b}\left({b}−\mathrm{2}{a}\right)} \\ $$$${r}={a}\sqrt{\frac{{h}}{{h}−\mathrm{2}{a}}} \\ $$$$\Rightarrow{r}={a}\sqrt{\frac{{b}+{a}+\sqrt{{b}\left({b}−\mathrm{2}{a}\right)}}{{b}−{a}+\sqrt{{b}\left({b}−\mathrm{2}{a}\right)}}} \\ $$

Commented by Tawa11 last updated on 23/Jan/22

Great sir

$$\mathrm{Great}\:\mathrm{sir} \\ $$

Commented by ajfour last updated on 24/Jan/22

Very straight n pragmatic,  really great sir. Thank you.

$${Very}\:{straight}\:{n}\:{pragmatic}, \\ $$$${really}\:{great}\:{sir}.\:{Thank}\:{you}. \\ $$

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