CBSE Class 11-science Answered

PQRS is a rectangle, PQ = a, QR = b

Let B be the center, r1 be the radius, draw a circle passing through P, Q and touching RS. Let E be the center, r2 be the radius, draw a circle passing through Q, R and touching PS.

BC ? RS and C is the midpoint of RS, extend CB to meet PQ at D.

  PD=DQ=a/2

  CD =b, BD+BC =b

   BD =b - r1 ---------(1)

    From triangle BDQ,    

 BD²=BQ²?DQ²

   BD = root [(r1) ²? a²/ 4] -----(2)

 From (1) and (2)

   b - r1 = root [(r1) ²? a²/ 4]

  Squaring both sides, we get

   b²?2br1+(r1)²=(r1) ²? a²/ 4

   2br1=b²+ a²/ 4

   r1=b/ 2+a² / 8b-------(3)

Similarly we get,

   r2=a/ 2+b² / 8a -------(4)

 From (3) and (4), we have

  r1+r2= a+b/ 2+a² + b²/ 8ab

 r1+ r2=a+b/ 2+(a+b)(a²?ab+b²/ 8ab)

 r1+r2=(a+b)(1/2+ a²?ab+b²/ 8ab)

 r1+r2=(a+b)(1/2+(a?b)²+ab / 8ab)

 r1+r2=(a+b)(1/2+(a?b)² /8ab+1/8)

 r1+r2=(a+b)(5/8+(a?b)² /8ab)

  r1+r2?5(a+b)/8

  8(r1+r2)?5(a+b)

Hence, 5(a+b)?8(r1+r2)