1I was going through my previous posts when I found out this one , a beauty . I tried to solve it another way , and goodness me , I found out a much simpler solution . Hope you like it -
Point X - { t , 1 } , i . e , center of the circle ;
Point A - { x , 1 / 2 } , whose x - co ordinate is to be found .
Point Y - { 0 , 1 / 2 } ,
Point O - { x , 0 } ,
XY = YZ = AO =1 / 2 { I am talking about distances }
XZ = 1 , AX = 1 { radius given 1 }
First , you should realize that the x - co ordinate , which we want to find , is ,
x = Length of arc AZ - Length of line AY ; ........................1
Now , from the triangle XYA ,
cos d = XY / AX = 1 / 2 ;
Hence , d = Î / 3 ;
Hence , length of arc AZ = R d { where R is the radius }
= { Î / 3 } . 1 ...............2
Again , sin d = AY / AX = AY ;
So , AY = √3 / 2 , ........................3
Hence , from 1 , 2 , and 3 ,
x = { Î / 3 } - { √3 / 2}
