Two circles C1, C2 are externally touching and have radii a, b. AB is their direct common tangent. A lies on C1 & B lies on C2. Find the radius of circle C with touches C1, C2 externally and also line AB
Answers
Given : Two circles C1 C2 externally touching and have radii a, b. AB is direct common tangent. A lies on C1 and B lies on C2 . circle C touches C1 and C2 externally and also line AB
To find : radius of circle C
Solution:
Refer the attached picture :
CX ║ AB
=> BX = a
& C₂X = b - a
C₁X = AB
C₁C₂ = a + b
(C₁C₂)² = C₁X² + C₂X²
(a + b)² = AB² + (b - a)²
=> a² + b² + 2ab = AB² + b² + a² - 2ab
=> AB² = 4ab
=> AB = 2√ab
Similarly
if we take Circle C radius as R
Then Taking C₁ & C Circles
AP = 2√aR
Taking C₂ & C Circles
BP = 2√bR
AB = AP + BP
=> 2√ab = 2√aR + 2√bR
Dividing by 2√abR
=> 1/√R = 1/√b + 1/ √a
=> 1/√R = (√a + √b)/√ab
=> √R = √ab / (√a + √b)
=> R = ab/ (√a + √b)²
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