બે લંબ રીતે સરળ આવર્ત ગતિ કરતાં પદાર્થ માટે સમીકરણ નીચે આપેલા છે.તેમને જોડીને Lissajous ગ્રાફ બનાવવામાં આવે છે. \(x\left( t \right) = A\,\sin \,\left( {at + \delta } \right)\) \(y\left( t \right) = B\,\sin \,\left( {bt} \right)\) તો નીચેનામાંથી શું સાચું પડશે?

A \(A\, = B\), \(a\, = 2b\); \(\delta = \frac{\pi }{2}\); ગ્રાફ : વર્તુળ
B \(A\,= B\), \(a\, = b\); \(\delta = \frac{\pi }{2}\) ; ગ્રાફ : રેખા
C \(A \ne B\), \(a\, = b\); \(\delta = \frac{\pi }{2}\); ગ્રાફ : ઉપવલય
D Parameters: \(A \ne B\), \(a\, = b\); \(\delta\, = 0\); ગ્રાફ : પરવલય

Verified Solution

Answer & Solution

Correct Answer

(C) \(A \ne B\), \(a\, = b\); \(\delta = \frac{\pi }{2}\); ગ્રાફ : ઉપવલય

Step-by-step Solution

Detailed explanation

From the two mutually perpendicular \(S.H.M. 's\), the general equation of Lissajous figure \(\frac{{{x^2}}}{{{A^2}}} + \frac{{{y^2}}}{{{B^2}}} - \frac{{2xy}}{{AB}}\cos \,\delta = {\sin ^2}\,\delta\) \(x = A\,\sin \,\left( {at + \delta } \right)\)…

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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