The circle x^2+y^2-2 x=0 is partitioned by line y=x in two segments. Let A ₁, ~A ₂ be the areas of major and minor segments respectively. What is the value of 2 (A₁+A₂ ) A₁-3 A₂ ?

Step-by-step solution

aligned Given, & & x^2+y^2-2 x & =0 & & (x-1)^2+y^2 & =1 or & & y^2 & =1-(x-1)^2 aligned Area of minor segment ( A ₂ ) aligned & = ₀^1 1-(x-1)^2-(x) d x ~A ₂ & = [ (x-1) 2 1-(x-1)^2 + 1 2 ⁻¹(x-1)- x^2 2 ]₀^1 ~A ₂ & = [ (0+ 1 2 (0)- 1 2 ) ]- [0+ 1 2 ⁻¹(-1)-0 ] A ₂ & = -2 4 aligned 2 ( ~A ₁+ A ₂ ) A ₁-3 ~A ₂ = 2( ) ( 3 +2 4 )- 3( -2) 4 =