The quarter circle with centre at O be x^2+y^2=R^2. T is then (RV3/2,R/2) where V = square root. Hence RT is represented by xRV3/2+yR/2=R^2 or xV3+y=2R. Considering point R, we can say, (OS)V3+R =2R or OS^2 = R^2/3 and OR^2=R^2+R^2/3=4R^2/3. Now let U be the foot of the perpendicular from T to OS. SM = OM - OS =RV3/2 -R/V3 = R/2V3 while TM=R/2. Hence,ST^2 = R^2/12 + R^2/4 = R^2/3. Finally, A(QOR)/A(QST)=(OR/ST)^2 since the triangles are similar. Hence the reqd. ratio= (4R^2/3)/(R^2/3) = 4 Ajit
The quarter circle with centre at O be x^2+y^2=R^2. T is then (RV3/2,R/2) where V = square root. Hence RT is represented by xRV3/2+yR/2=R^2 or xV3+y=2R. Considering point R, we can say, (OS)V3+R =2R or OS^2 = R^2/3 and OR^2=R^2+R^2/3=4R^2/3. Now let U be the foot of the perpendicular from T to OS. SM = OM - OS =RV3/2 -R/V3 = R/2V3 while TM=R/2. Hence,ST^2 = R^2/12 + R^2/4 = R^2/3. Finally, A(QOR)/A(QST)=(OR/ST)^2 since the triangles are similar. Hence the reqd. ratio= (4R^2/3)/(R^2/3) = 4
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Erratum:
ReplyDeleteLet M be the foot of the perpendicular from T to OS
Ajit