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Question Number 74698 by ajfour last updated on 29/Nov/19

Commented by ajfour last updated on 29/Nov/19

If AB^(⌢) =AE^(⌢) = DE^(⌢) . Find equation of circle.

$${If}\:\overset{\frown} {{AB}}\:=\overset{\frown} {{AE}}\:=\:\overset{\frown} {{DE}}\:. \\ $$$${Find}\:{equation}\:{of}\:{circle}. \\ $$

Answered by ajfour last updated on 29/Nov/19

Let C(0,c) . ∠DCE = 60° . let radius be r. E(((r(√3))/2), c+(r/2)) c+(r/2)= ((3r^2 )/4) .....(i) c = rsin 30° = (r/2) ....(ii) ⇒ r = (4/3) , c = (2/3) . Eq. of circle : x^2 +(y−(2/3))^2 = ((16)/3) or 9x^2 +9y^2 −12y−44 = 0 .

$${Let}\:\:{C}\left(\mathrm{0},{c}\right)\:.\:\:\angle{DCE}\:=\:\mathrm{60}°\:. \\ $$$${let}\:{radius}\:{be}\:{r}. \\ $$$${E}\left(\frac{{r}\sqrt{\mathrm{3}}}{\mathrm{2}},\:{c}+\frac{{r}}{\mathrm{2}}\right) \\ $$$${c}+\frac{{r}}{\mathrm{2}}=\:\frac{\mathrm{3}{r}^{\mathrm{2}} }{\mathrm{4}}\:\:\:\:\:\:\:\:\:\:\:.....\left({i}\right) \\ $$$${c}\:=\:{r}\mathrm{sin}\:\mathrm{30}°\:=\:\frac{{r}}{\mathrm{2}}\:\:\:\:....\left({ii}\right) \\ $$$$\Rightarrow\:\:{r}\:=\:\frac{\mathrm{4}}{\mathrm{3}}\:,\:\:{c}\:=\:\frac{\mathrm{2}}{\mathrm{3}}\:. \\ $$$${Eq}.\:{of}\:{circle}\:: \\ $$$$\:\:{x}^{\mathrm{2}} +\left({y}−\frac{\mathrm{2}}{\mathrm{3}}\right)^{\mathrm{2}} \:=\:\frac{\mathrm{16}}{\mathrm{3}} \\ $$$${or}\:\:\:\:\:\mathrm{9}{x}^{\mathrm{2}} +\mathrm{9}{y}^{\mathrm{2}} −\mathrm{12}{y}−\mathrm{44}\:=\:\mathrm{0}\:. \\ $$

Answered by mind is power last updated on 29/Nov/19

⇒∠DCE=∠ECA=∠ACD=(π/3) OC=AC cos((π/3))=(R/2) C(0,(R/2)) E=(Rcos((π/6)),Rsin((π/6))+(R/2)) E(R((√3)/2),R)∈y=x^2 ⇒(R((√3)/2))^2 =R⇒R(1−R.(3/4))=0⇒R=(4/3)

$$\Rightarrow\angle\mathrm{DCE}=\angle\mathrm{ECA}=\angle\mathrm{ACD}=\frac{\pi}{\mathrm{3}} \\ $$$$\mathrm{OC}=\mathrm{AC}\:\mathrm{cos}\left(\frac{\pi}{\mathrm{3}}\right)=\frac{\mathrm{R}}{\mathrm{2}} \\ $$$$\mathrm{C}\left(\mathrm{0},\frac{\mathrm{R}}{\mathrm{2}}\right) \\ $$$$\mathrm{E}=\left(\mathrm{Rcos}\left(\frac{\pi}{\mathrm{6}}\right),\mathrm{Rsin}\left(\frac{\pi}{\mathrm{6}}\right)+\frac{\mathrm{R}}{\mathrm{2}}\right) \\ $$$$\mathrm{E}\left(\mathrm{R}\frac{\sqrt{\mathrm{3}}}{\mathrm{2}},\mathrm{R}\right)\in\mathrm{y}=\mathrm{x}^{\mathrm{2}} \Rightarrow\left(\mathrm{R}\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\right)^{\mathrm{2}} =\mathrm{R}\Rightarrow\mathrm{R}\left(\mathrm{1}−\mathrm{R}.\frac{\mathrm{3}}{\mathrm{4}}\right)=\mathrm{0}\Rightarrow\mathrm{R}=\frac{\mathrm{4}}{\mathrm{3}} \\ $$$$ \\ $$$$ \\ $$

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