Ex 5.6, 2 - Chapter 5 Class 12 Continuity and Differentiability

Transcript

Ex 5.6, 2 If x and y are connected parametrically by the equations without eliminating the parameter, Find 𝑑𝑦/𝑑𝑥, 𝑥=𝑎 cos⁡𝜃, 𝑦=𝑏 cos⁡𝜃 𝑥=𝑎 cos⁡𝜃, 𝑦=𝑏 cos⁡𝜃 𝑑𝑦/𝑑𝑥 = 𝑑𝑦/𝑑𝑥 × 𝑑𝜃/𝑑𝜃 (█("Multiplying Numerator" @"& Denominator by d" 𝜃)) 𝑑𝑦/𝑑𝑥 = 𝑑𝑦/𝑑𝜃 × 𝑑𝜃/𝑑𝑥 𝑑𝑦/𝑑𝑥 = (𝑑𝑦/𝑑𝜃)/(𝑑𝑥/𝑑𝜃) Calculating 𝒅𝒚/𝒅𝜽 𝑦 = 𝑏 cos⁡𝜃 𝑑𝑦/𝑑𝜃 = 𝑑(𝑏 𝑐𝑜𝑠⁡𝜃)/𝑑𝜃 𝑑𝑦/𝑑𝜃 = b . (−sin⁡𝜃 ) 𝑑𝑦/𝑑𝜃 = − b sin 𝜃 Calculating 𝒅𝒙/𝒅𝜽 𝑥=𝑎 cos⁡𝜃 𝑑𝑥/𝑑𝜃 = 𝑑(𝑎 cos⁡𝜃 )/𝑑𝜃 𝑑𝑥/𝑑𝜃 = 𝑎 . (−sin⁡𝜃 ) 𝑑𝑥/𝑑𝜃 = −𝑎 sin⁡𝜃 Therefore 𝑑𝑦/𝑑𝑥 = (𝑑𝑦/𝑑𝜃)/(𝑑𝑥/𝑑𝜃) 𝑑𝑦/𝑑𝑥 = (− b sin 𝜃)/(− 𝑎 sin⁡𝜃 ) 𝒅𝒚/𝒅𝒙 = 𝒃/𝒂