Question Number 102524 by ajfour last updated on 09/Jul/20

Commented byajfour last updated on 09/Jul/20

Given three circles of radii p, q, r touching the coordinate axes as shown; find equation of the circum- circle.

Answered by mr W last updated on 09/Jul/20

center (h, k) radius R (h−p)^2 +(k−p)^2 =(R−p)^2 ...(i) (h+q)^2 +(k−q)^2 =(R−q)^2 ...(ii) (h−r)^2 +(k+r)^2 =(R−r)^2 ...(iii) (i)−(ii): (2h−p+q)(−p−q)+(2k−p−q)(−p+q)=(2R−p−q)(−p+q) ⇒((p+q)/(p−q))h+k=R+((p+q)/2) ...(I) (i)−(iii): (2h−p−r)(p−r)+(2k−p+r)(p+r)=(2R−p−r)(p−r) ⇒h+((p+r)/(p−r))k=R+((p+r)/2) ...(II) ⇒h=(((R+((p+q)/2))(((p+r)/(p−r)))−(R+((p+r)/2)))/((((p+q)/(p−q)))(((p+r)/(p−r)))−1)) ⇒k=(((R+((p+r)/2))(((p+q)/(p−q)))−(R+((p+q)/2)))/((((p+q)/(p−q)))(((p+r)/(p−r)))−1)) we put this into (i) to get R...

Commented bymr W last updated on 09/Jul/20

Commented byajfour last updated on 10/Jul/20

Straight enough Sir, thanks a lot, i have no better way..