in the adjoining figure angle ACE is a right triangle there are three circles which just touch each other and AC and EC are the tangents to all the three circles .What is the ratio of radii of the largest circle to that of the smallest circle
Answers
Given : ACE is a right angle triangle
To find : ratio of radii of the largest circle to that of the smallest circle
Solution:
Let say Distance of C from outer of circle is d
and radius of smallest circle = r₁
∠ACE = 90°
=> half of angle = 45°
Now triangle formed with center of smallest circle center , tangent & point d has two angles 45° & one angle 90°
=> r₁² + r₁² = (r₁ + d)²
=> r₁ + d = √2 r₁
=> d = r₁ (√2 - 1)
Now Similarly
r₂² + r₂² = (r₂ + 2r₁ + d)²
=> 2r₂² = (r₂ + 2r₁ + r₁ (√2 - 1))²
=> √2r₂ = r₂ + 2r₁ + r₁ (√2 - 1)
=> r₂ (√2 - 1) = r₁ (√2 + 1)
=> r₂ = r₁ (√2 + 1) / (√2 - 1)
=> r₂ = r₁ (2 + 1 + 2√2)
=> r₂ = r₁ (3 + 2√2)
now
r₃² + r₃² = ( r₃ + 2r₂ + 2r₁ + d)²
=> 2 r₃² = ( r₃ + 2r₁ (3 + 2√2) + 2r₁ + r₁ (√2 - 1) )²
=> √2 r₃ = r₃ + r₁ (6 + 4√2 + 2 + √2 - 1 )
=> r₃ (√2 - 1) = r₁ ( 7 + 5√2)
=> r₃ / r₁ = ( 7 + 5√2) / (√2 - 1)
=> r₃ / r₁ = ( 7 + 5√2) (√2 + 1)
=> r₃ / r₁ = 7√2 + 10 + 7 + 5√2
=> r₃ / r₁ = 17 + 12√2
ratio of radii of the largest circle to that of the smallest circle = 17 + 12√2
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