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NCERT Class 12 · Ch 5 Exercise 5.6 · Q9

NCERT Class 12 Maths · Chapter 5 Continuity And Differentiability · Exercise 5.6 Question 9

Without eliminating parameter t (θ in the trigonometric source forms), find dy/dx for x=\(a \sec{\left(t \right)}\), y=\(b \tan{\left(t \right)}\).

💡 Concept Hints

  1. Identify the domain and the inner and outer functions, or the relevant one-sided limits.
  2. Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.

✅ Step-by-Step Solution

Final Answer: \(\frac{b}{a \sin{\left(t \right)}}\) where the real parametrisation is defined and dx/dt≠0.
  1. Differentiate both coordinates: dx/dt=\(a \tan{\left(t \right)} \sec{\left(t \right)}\), dy/dt=\(b \left(\tan^{2}{\left(t \right)} + 1\right)\).
  2. Divide the two rates: dy/dx=(dy/dt)/(dx/dt)=\(\frac{b \left(\tan^{2}{\left(t \right)} + 1\right)}{a \tan{\left(t \right)} \sec{\left(t \right)}}\).
  3. Simplify to \(\frac{b}{a \sin{\left(t \right)}}\). This quotient requires dx/dt≠0; any cancelled zero requires a separate limiting or curve check.

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