ધારો કે વર્તૂળ \(C\) નું કેન્દ્ર \((1,1)\) અને ત્રિજ્યા \( 1\) છે.જો \( (0,y)\) કેન્દ્રવાળું વર્તૂળ \(T \) ઊગમબિંદુમાંથી પસાર થતું હોય અને વર્તૂળ \(C \) ને બહારથી સ્પર્શતું હોય તો વર્તૂળ \(T \) ની ત્રિજ્યા મેળવો.

A \(\frac{1}{2}\)
B \(\frac{1}{4}\)
C \(\frac{{\sqrt 3 }}{{\sqrt 2 }}\)
D \(\frac{5}{4}\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\frac{1}{4}\)

Step-by-step Solution

Detailed explanation

\(C T^{2}=(1-0)^{2}+(1-y)^{2}\) also \(C T=1+y\) \(\therefore(1+y)^{2}=1+(1-y)^{2}\) \(1^{2}+2 y+y^{2}=1+1^{2}-2 y+y^{2}\) \(2 y+2 y=1+1^{2}+y^{2}-1^{2}-y^{2}\) \(4 y=1\) \(\Longrightarrow y=\frac{1}{4}\) Hence the radius of \(T\) is \(\frac{1}{4}\)

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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