Ex 5.6, 9 - Chapter 5 Class 12 Continuity and Differentiability

Transcript

Ex 5.6, 9 If x and y are connected parametrically by the equations without eliminating the parameter, Find 𝑑𝑦﷮𝑑𝑥﷯, 𝑥=𝑎 sec ⁡𝜃, 𝑦=𝑏 tan⁡𝜃 𝑥=𝑎 sec ⁡𝜃, 𝑦=𝑏 tan⁡𝜃 𝑑𝑦﷮𝑑𝑥﷯ = 𝑑𝑦﷮𝑑𝑥﷯ × 𝑑𝜃﷮𝑑𝜃﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑑𝑦﷮𝑑𝜃﷯ × 𝑑𝜃﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑑𝑦﷮𝑑𝜃﷯﷮ 𝑑𝑥﷮𝑑𝜃﷯﷯ Calculating 𝒅𝒚﷮𝒅𝜽﷯ 𝑦 = 𝑏 tan﷮𝜃﷯ 𝑑𝑦﷮𝑑𝜃﷯ = 𝑑 𝑏 tan﷮𝜃﷯﷯﷮𝑑𝜃﷯ 𝑑𝑦﷮𝑑𝜃﷯ = 𝑏 𝑑 tan﷮𝜃﷯﷯﷮𝑑𝜃﷯ 𝑑𝑦﷮𝑑𝜃﷯ = 𝑏 . sec﷮2﷯﷮𝜃﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑏 . 1﷮ cos﷮𝜃﷯﷯﷮𝑎 sin﷮𝜃﷯﷮ cos﷮𝜃﷯﷯﷯﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑏 × 1﷮ cos﷮𝜃﷯﷯ × cos﷮𝜃﷯﷮ a sin﷮𝜃﷯﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑏﷮𝑎﷯ 1﷮ sin﷮𝜃﷯﷯﷯ 𝒅𝒚﷮𝒅𝒙﷯ = 𝒃﷮𝒂﷯ 𝒄𝒐𝒔𝒆𝒄 𝜽