Solve this also 52 Three Circles with centres C1, C2, C3 and radii r1,r2, r3, where r1 r2 - Maths - Triangles

Question

Solve this also 52
Three Circles with centres C1, C2,
C3 and radii r1,r2 ,
r3, where r1<r2 <
r3 are placed as shown in the figure

Then r2 = __________
(1) ( 1 )   r 3 - r 1          
                 
        ( 2 )   r 3 + r 1    
                 
              ( 3 )   r 3 r
1                  
                 
  ( 4 )   r 3 2 - r 1 2

Shivam Mishra · Accepted Answer

Dear Student,
Please find below the solution to the asked query:

<img alt="" src=
"https://s3mn%20mnimgs%20com/img/shared/content_ck_images/ck_59d3bb81441af%20jpg"
style="width: 501px; height: 208px;">
In ∠PC1A,∠PC2B,∠PC3Cone angle is 90° and angle P is common, hence∆PC1A,∆PC2B,∆PC3C will be similar
Hence for ∆PC1A and ∆PC2BPAr1=PBr2⇒PAPB=r1r2
iFor ∆PC2B and ∆PC3C⇒PBr2=PCr3⇒PBPC=r2r3
iiBut  we know that if tangents are drawn from an external point Pto three circles which touch the circle at A, B and  C respectively, then  PAPB=PBPC  Use i and ii  r1r2= r2r3    ⇒r22=r1r3⇒r2=r1r3

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Solve this also 52. Three Circles with centres C1, C2, C3 and radii r1,r2 , r3, where r1<r2 < r3 are placed as shown in the figure. Then r2 = __________ (1) ( 1 ) r 3 - r 1 ( 2 ) r 3 + r 1 ( 3 ) r 3 r 1 ( 4 ) r 3 2 - r 1 2

Solve this also 52.
Three Circles with centres C₁, C₂, C₃ and radii r₁,r₂ _, r₃, where r₁<r₂ < r₃ are placed as shown in the figure.

Then r₂ = __________
(1) $(1) \sqrt{r_{3} - r_{1}} (2) \sqrt{r_{3} + r_{1}} (3) \sqrt{r_{3} r_{1}} (4) \sqrt{{r_{3}}^{2} - {r_{1}}^{2}}$