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Three circles touch the one another externally. The tangents at their point of contact meet at a point whose distance from a point of contanct is 4. Find the product of the radii to the sum of the radii of the circles

Three circles touch the one another externally. The tangents at their point of contact meet at a point whose distance from a point of contanct is 4. Find the product of the radii to the sum of the radii of the circles

Grade:11

1 Answers

Jitender Pal
askIITians Faculty 365 Points
6 years ago
Hello Student,
Please find the answer to your question
Let us consider circles with centers at A, B and C and with radii r1, r2and r3respectively which touch each other externally at P, Q and R. Let the common tangents at P, Q and R meet each other at O. Then OP = OQ = OR = 4 (given)(lengths of tangents from a pt to a circle are equal).
Also OP ⊥ AB, OQ ⊥ AC, OR ⊥BC.
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⇒ O is the in centre of the ∆ABC Thus for ∆ ABC
s = (r1 + r2) + (r2 + r3) + (r3 + r1)/2
i.e. s = (r1 + r2 + r3)
∴ ∆ = √(r1 + r2 + r3) r1 r2 r3 (Heron’s formula)
Now r = ∆/s
NOTE THIS STEP :
⇒ 4 = √(r1 + r2 + r3) r1 r2 r3/r1 + r2 + r3
⇒ 4 = √r1 r2 r3/√r1 + r2 + r3
⇒ r1 r2 r3/r1 + r2 r3 = 16/1
⇒ r1. r2. r3 : r1 + r2 + r3 = 16 : 1

Thanks
Jitender Pal
askIITians Faculty

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