વર્તુળ \(C_{1}\) એ ઉગમબિંદુ \(O\) માંથી પસાર થાય છે અને ધન \(x-\) અક્ષ પર \(4\) લંબાઇનો વ્યાસ છે. રેખા \(y =2 x\) એ વર્તુળ \(C _{1}\) પર જીવા \(OA\) બનાવે છે. અહી \(C _{2}\) માં \(OA\) વ્યાસ છે. જો \(C _{2}\) નો બિંદુ \(A\) આગળનો સ્પર્શક \(x\)-અક્ષને બિંદુ \(P\) અને \(y\)-અક્ષને \(Q\) માં છેદે છે તો \(QA : AP\) ની કિમંત મેળવો.

A \(1:4\)
B \(1: 5\)
C \(2: 5\)
D \(1: 3\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(1:4\)

Step-by-step Solution

Detailed explanation

\(C _{2}\) is a circle with \(OA\) as diameter. So, tangent at \(A\) on \(C _{2}\) is perpendicular to \(OR\) Let \(OA =\ell\) \(\therefore \frac{ QA }{ AP }=\frac{\ell \cot \theta}{\ell \tan \theta}\) \(=\frac{1}{\tan ^{2} \theta}=\frac{1}{4}\)

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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