NTA Abhyas JEE Main2020MathematicsCirclePractice

Let four circles having radii r 1 = 5 units , r 2 = 5 units , r 3 = 8 units and r 4 units r 4 < 5 are mutually touching each other externally, then the value of 2 r 4 is equal to

Correct answer

2.25

Step-by-step solution

Let, the centres of circles with radii r 1 , r 2 , r 3 , r 4 are C 1 , C 2 , C 3 and C 4 respectively. The foot of C 3 on C 1 C 2 is E , then C 3 E = 1 3 2 - 5 2 = 12 , C 4 E = 12 - 8 - r 4 = 4 - r 4 . Now, C 4 C 2 2 = C 4 E 2 + 5 2 ⇒ 5 + r 4 2 = 4 - r 4 2 + 5 2 ⇒ r 4 = 8 9 ⇒ 2 r 4 = 9 4 = 2 .25

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