The-circle-touches-the-circle-internally-at-P-The-centre-O-of-is-outside-Let-XY-be-a-diameter-of-which-is-also-tangent-to-Assume-PY-gt-PX-Let-PY-intersect-at-Z-If-YZ-2PZ-what-i

Question Number 17524 by 786 last updated on 07/Jul/17

$$\mathrm{The}\:\mathrm{circle}\:\omega\:\mathrm{touches}\:\mathrm{the}\:\mathrm{circle}\:\Omega \\ $$$$\mathrm{internally}\:\mathrm{at}\:{P}.\:\mathrm{The}\:\mathrm{centre}\:{O}\:\mathrm{of}\:\Omega\:\mathrm{is} \\ $$$$\mathrm{outside}\:\omega.\:\mathrm{Let}\:{XY}\:\mathrm{be}\:\mathrm{a}\:\mathrm{diameter}\:\mathrm{of}\:\Omega \\ $$$$\mathrm{which}\:\mathrm{is}\:\mathrm{also}\:\mathrm{tangent}\:\mathrm{to}\:\omega.\:\mathrm{Assume} \\ $$$${PY}\:>\:{PX}.\:\mathrm{Let}\:{PY}\:\mathrm{intersect}\:\omega\:\mathrm{at}\:{Z}.\:\mathrm{If} \\ $$$${YZ}\:=\:\mathrm{2}{PZ},\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{magnitude}\:\mathrm{of} \\ $$$$\angle{PYX}\:\mathrm{in}\:\mathrm{degrees}? \\ $$

Answered by ajfour last updated on 07/Jul/17

$$\angle\mathrm{PYX}=\mathrm{15}°\:. \\ $$

Commented by ajfour last updated on 07/Jul/17

Commented by ajfour last updated on 08/Jul/17

$$\mathrm{YZ}=\mathrm{2PZ}\:\:\:\mathrm{or}\:\:\mathrm{3PZ}=\mathrm{PY} \\ $$$$\mathrm{3}\left(\mathrm{2rcos}\:\theta\right)=\mathrm{2Rcos}\:\theta \\ $$$$\Rightarrow\:\:\:\:\mathrm{R}=\mathrm{3r} \\ $$$$\:\:\mathrm{In}\:\bigtriangleup\mathrm{OCN},\:\mathrm{sin}\:\mathrm{2}\theta=\frac{\mathrm{r}}{\mathrm{R}−\mathrm{r}}\:=\frac{\mathrm{r}}{\mathrm{3r}−\mathrm{r}} \\ $$$$\:\:\:\:\:\mathrm{sin}\:\mathrm{2}\theta\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:\:\:\mathrm{or}\:\:\:\:\theta=\mathrm{15}°\:. \\ $$

Commented by Tinkutara last updated on 07/Jul/17

$$\mathrm{Thanks}\:\mathrm{Sir}! \\ $$

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