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Question Number 190405 by cortano12 last updated on 02/Apr/23

Answered by cherokeesay last updated on 02/Apr/23

Answered by mr W last updated on 02/Apr/23

((p−q)/(p+q))=((q−r)/(q+r))=k  q=((1+k)/(1−k))r  p=((1+k)/(1−k))q=(((1+k)/(1−k)))^2 r  (p+q)^2 −(p−q)^2 =a^2   4pq=a^2   4(((1+k)/(1−k)))^3 r^2 =a^2    ...(i)  (q+r)^2 −(q−r)^2 =b^2   4qr=b^2   4(((1+k)/(1−k)))r^2 =b^2    ...(ii)  ⇒((1+k)/(1−k))=(a/b)  4×(a/b)×r^2 =b^2   ⇒r=(b/2)(√(b/a))  ⇒q=(b/2)((a/b))(√(b/a))  ⇒p=(b/2)((a/b))^2 (√(b/a))  R=p+q+r=[((a/b))^2 +((a/b))+1](b/2)(√(b/a))  A_(red) =(π/2)(R^2 −p^2 −q^2 −r^2 )      =(π/2){[((a/b))^2 +((a/b))+1]^2 −((a/b))^4 −((a/b))^2 −1}(b^2 /4)×(b/a)      =π{((a/b))^3 +((a/b))^2 +((a/b))}(b^2 /4)×(b/a)      ={((a/b))^2 +(a/b)+1}((πb^2 )/4)      ={((8/4))^2 +(8/4)+1}((π×4^2 )/4)      =28π

pqp+q=qrq+r=kq=1+k1krp=1+k1kq=(1+k1k)2r(p+q)2(pq)2=a24pq=a24(1+k1k)3r2=a2...(i)(q+r)2(qr)2=b24qr=b24(1+k1k)r2=b2...(ii)1+k1k=ab4×ab×r2=b2r=b2baq=b2(ab)bap=b2(ab)2baR=p+q+r=[(ab)2+(ab)+1]b2baAred=π2(R2p2q2r2)=π2{[(ab)2+(ab)+1]2(ab)4(ab)21}b24×ba=π{(ab)3+(ab)2+(ab)}b24×ba={(ab)2+ab+1}πb24={(84)2+84+1}π×424=28π

Commented by mr W last updated on 02/Apr/23

Answered by a.lgnaoui last updated on 03/Apr/23

Soit :C_1 (O1,R_1 ) ;   C_2 (O_2 ,R_2 )   ;C_3 (O_3 ,R_3 )  R_1 =O_1 C     R_2 =O_2 B     R_3 =O_3 A  1•tan θ=((r_2 −r_1 )/4)=((r_3 −r_2 )/8)=((r_3 −r_1 )/(12))(1)  2• { ((4^2 +_1 (r_2 −r_1 )^2 =(r_1 +r_2 )^2       (2))),((8^2 +(r_3 −r_2 )^2 =(r_3 +r_2 )^2         (3))) :}  (1) ⇒ { ((2(r_2 −r_1 )=(r_3 −r_2 ))),((3(r_3 −r_2 )=2(r_3 −r_1 ))) :}     ⇒       r_3 =3r_2 −2r_1         (4)  (2) { ((r_1 r_2 =4                     (5))),((r_2 r_3 =16                   (6))) :}  (5)et(6)⇒  r_3 =4r_1    (7)  (4)⇒4r_1 =3r_2 −2r_1          ⇒                  r_2 =2r_1     (8)    3•   (1)⇒12^2 +(r_3 −r_1 )^2 =(r_1 +2r_2 +r_3 )^2     36=r_2 ^2 +r_1 r_3 +r_2 (r_1 +r_3 )  =4r_1 ^2 +4r_1 ^2 +2r_1 (5r_1 )=18r_1 ^2   36=18r_1 ^2      ⇒r_1 =(√2)   Rayon R=r_1 +2r_1 +4r_1 =7r_1   ⇒       R=7(√2)  Aire totale=πR^2   Aire (Rouge)=((πR^2 )/2)−π(((r_1 ^2 +r_2 ^2 +r_3 ^2 )/2))           r_1 ^2 +r_2 ^2 +r_3 ^2 =42   { ((Aire=28π)),((Aire   ≅87,96)) :}

Soit:C1(O1,R1);C2(O2,R2);C3(O3,R3)R1=O1CR2=O2BR3=O3A1tanθ=r2r14=r3r28=r3r112(1)2{42+1(r2r1)2=(r1+r2)2(2)82+(r3r2)2=(r3+r2)2(3)(1){2(r2r1)=(r3r2)3(r3r2)=2(r3r1)r3=3r22r1(4)(2){r1r2=4(5)r2r3=16(6)(5)et(6)r3=4r1(7)(4)4r1=3r22r1r2=2r1(8)3(1)122+(r3r1)2=(r1+2r2+r3)236=r22+r1r3+r2(r1+r3)=4r12+4r12+2r1(5r1)=18r1236=18r12r1=2RayonR=r1+2r1+4r1=7r1R=72Airetotale=πR2Aire(Rouge)=πR22π(r12+r22+r322)r12+r22+r32=42{Aire=28πAire87,96

Answered by Skabetix last updated on 10/Apr/23

Commented by Skabetix last updated on 10/Apr/23

Answered by ajfour last updated on 10/Apr/23

2(√(pq))=a  2(√(qr))=b  (p/q)=(q/r)  ⇒  4rp=4q^2 =ab  R=p+q+r  ((2S)/π)=(p+q+r)^2 −p^2 −q^2 −r^2         =2(pq+qr+rp)       =2((a^2 /4)+(b^2 /4)+((ab)/4))  ⇒ S=(π/4)(a^2 +b^2 +ab)           =(π/4)(64+16+32)=28π

2pq=a2qr=bpq=qr4rp=4q2=abR=p+q+r2Sπ=(p+q+r)2p2q2r2=2(pq+qr+rp)=2(a24+b24+ab4)S=π4(a2+b2+ab)=π4(64+16+32)=28π

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