Question Number 71348 by ajfour last updated on 13/Oct/19

Commented byajfour last updated on 13/Oct/19

If the circle has unit radius and each part have same area, find radius of circular arc.

Commented byajfour last updated on 14/Oct/19

Commented byajfour last updated on 14/Oct/19

let center of small circle be origin. p^2 +q^2 =1 p^2 +(q+1+k)^2 =R^2 ⇒ (1+k)(2q+1+k)=R^2 −1 ⇒ q=(1/2)[((R^2 −1)/(1+k))−(1+k)] < 0 ⇒ p=(√(1−(1/4)[((R^2 −1)/(1+k))−(1+k)]^2 )) let θ=tan^(−1) ((∣q∣)/p) A_1 =(((π/2−θ))/2) ; let β=tan^(−1) (p/(q+1+k)) A_2 =((R^2 β)/2) & A_3 =((p(1+k))/2) Now A_1 −(A_2 −A_3 )=(π/4) ⇒ (((π/2−θ))/2)−((R^2 β)/2)+((p(1+k))/2)=(π/4) A_4 =(R^2 /2)cos^(−1) (k/R)−((k(√(R^2 −k^2 )))/2)=(π/2) θ=tan^(−1) (((−(1/2)[((R^2 −1)/(1+k))−(1+k)])/(√(1−(1/4)[((R^2 −1)/(1+k))−(1+k)]^2 )))) β=tan^(−1) (p/(q+1+k)) __________________________.