Ex 9.3, 6

Transcript

Ex 9.3, 6 Form the differential equation of the family of circle touching the 𝑦−𝑎𝑥𝑖𝑠 at origin. General Equation of Circle 𝑥−𝑎﷯﷮2﷯+ 𝑦−𝑏﷯﷮2﷯= 𝑟﷮2﷯ where Centre At 𝑎 , 𝑏﷯ and Radius is r If circle touches y-axis at origin, Center will be at x-axis So, Center = (a, 0) & Radius = a Thus, equation of circle becomes 𝑥−𝑎﷯﷮2﷯+ 𝑦−0﷯﷮2﷯= 𝑎﷮2﷯ 𝑥−𝑎﷯﷮2﷯+ 𝑦﷮2﷯= 𝑎﷮2﷯ 𝑥﷮2﷯+ 𝑐﷮2﷯−2𝑎𝑥+ 𝑦﷮2﷯= 𝑎﷮2﷯ 𝑥﷮2﷯−2𝑎𝑥+ 𝑦﷮2﷯= 𝑎﷮2﷯− 𝑎﷮2﷯ 𝑥﷮2﷯−2𝑎𝑥+ 𝑦﷮2﷯=0 2𝑎𝑥= 𝑥﷮2﷯+ 𝑦﷮2﷯ 2𝑎= 𝑥﷮2﷯ + 𝑦﷮2﷯﷮𝑥﷯ 2𝑎= 𝑥﷮2﷯﷮𝑥﷯+ 𝑦﷮2﷯﷮𝑥﷯ 2𝑎=𝑥+ 𝑦﷮2﷯﷮𝑥﷯ Differentiating Both Sides w.r.t. 𝑥 0=1+ 𝑑 𝑦﷮2﷯﷯﷮𝑑𝑥﷯ . 𝑥 − 𝑦﷮2﷯ 𝑑𝑥﷮𝑑𝑥﷯﷮ 𝑥﷮2﷯﷯ 0=1+ 2𝑦 . 𝑑𝑦﷮𝑑𝑥﷯ . 𝑥− 𝑦﷮2﷯﷮ 𝑥﷮2﷯﷯ −1= 2𝑥𝑦 𝑑𝑦﷮𝑑𝑥﷯ − 𝑦﷮2﷯﷮ 𝑥﷮2﷯﷯ −1. 𝑥﷮2﷯=2𝑥𝑦 𝑑𝑦﷮𝑑𝑥﷯− 𝑦﷮2﷯ − 𝑥﷮2﷯=2𝑥𝑦 𝑑𝑦﷮𝑑𝑥﷯− 𝑦﷮2﷯ 2𝑥𝑦 . 𝑦﷮′﷯﷯= 𝑦﷮2﷯− 𝑥﷮2﷯ 𝟐𝒙𝒚 𝒚﷮′﷯+ 𝒙﷮𝟐﷯= 𝒚﷮𝟐﷯ is the required differential equation.