ત્રણ વર્તુળ જેમની ત્રિજ્યા અનુક્રમે \(a, b, c\, ( a < b < c )\) છે તે એકબીજાને બહારથી સ્પર્શે છે જો તેમનો સામાન્ય સ્પર્શક \(x -\) અક્ષ હોય તો

A \(\frac{1}{{\sqrt a }} = \frac{1}{{\sqrt b }} + \frac{1}{{\sqrt c }}\)
B \(\frac{1}{{\sqrt b }} = \frac{1}{{\sqrt a }} + \frac{1}{{\sqrt c }}\)
C \(a, b, c\) સમાંતર શ્રેણીમાં છે
D \(\sqrt a ,\sqrt b ,\sqrt c \) સમાંતર શ્રેણીમાં છે

Verified Solution

Answer & Solution

Correct Answer

(A) \(\frac{1}{{\sqrt a }} = \frac{1}{{\sqrt b }} + \frac{1}{{\sqrt c }}\)

Step-by-step Solution

Detailed explanation

Length of direct common tangent for circle \({C_1}\) and \({C_2}\) is \(AB = \sqrt {{{\left( {a + b} \right)}^2} - {{\left( {a - b} \right)}^2}} \) For \({C_3}\) and \({C_2}\) Length of direct common tangent for is…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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