Question Number 105001 by ajfour last updated on 25/Jul/20

Commented byajfour last updated on 25/Jul/20

If both circles have unit radius, and  regions 1, 2, 3, 4, 5 have equal areas,  find eq. of both circles.

Answered by ajfour last updated on 25/Jul/20

let lower circle eq. be      (x−h)^2 +(y+k)^2 =1  upper circle eq. is then      (x+k)^2 +(y−h)^2 =1  let A(a,0)   & B(b,0)   let  x_0 =a, b  ⇒  x_0 =h±(√(1−k^2 ))  ⇒   a=h+(√(1−k^2 ))   ,   b=h−(√(1−k^2 ))   P (p,p) lies on y=x and both circles  hence   (p+k)^2 +(p−h)^2 =1  A_5 = ∫_b ^( 0) (−k+(√(1−(x−h)^2 )) )dx  A_2 +A_3  = ∫_0 ^(  a) (−k+(√(1−(x−h)^2 )) )dx  A_3  = 2∫_0 ^( p) (−k+(√(1−(x−h)^2 ))−x)dx  Now    A_2 +A_3  = 2A_5    ....(i)  &                 A_3  = A_5               ....(ii)  .....