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Class 12 Continuity And Differentiability · Short Answer2 Marksparametric_derivatives

Class 12 Maths · Continuity And Differentiability · Short Answer #57

For x=\(a \cos{\left(t \right)}\), y=\(b \sin{\left(t \right)}\), find d²y/dx². Assume a≠0 and sin t≠0.

✅ Step-by-Step Solution

Final Answer: \(- \frac{b}{a^{2} \sin^{3}{\left(t \right)}}\)
  1. Differentiate in t: dx/dt=\(- a \sin{\left(t \right)}\) and dy/dt=\(b \cos{\left(t \right)}\).
  2. Thus dy/dx=(dy/dt)/(dx/dt)=\(- \frac{b}{a \tan{\left(t \right)}}\). Applying d/dx=(1/(dx/dt))d/dt gives d²y/dx²=\(- \frac{b}{a^{2} \sin^{3}{\left(t \right)}}\).
  3. Therefore d²y/dx²=\(- \frac{b}{a^{2} \sin^{3}{\left(t \right)}}\).

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Differentiate in t: dx/dt=\(- a \sin{\left(t \right)}\) and dy/dt=\(b \cos{\left(t \right)}\).
  • [1 mark] Thus dy/dx=(dy/dt)/(dx/dt)=\(- \frac{b}{a \tan{\left(t \right)}}\). Applying d/dx=(1/(dx/dt))d/dt gives d²y/dx²=\(- \frac{b}{a^{2} \sin^{3}{\left(t \right)}}\).
  • [0.5 marks] Therefore d²y/dx²=\(- \frac{b}{a^{2} \sin^{3}{\left(t \right)}}\).

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