Ex 9.3, 10 - Family of circle having center on y-axis - Ex 9.3

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Ex 9.3, 10 Form the differential equation of the family of circle having a center on 𝑦−𝑎𝑥𝑖𝑠 and radius 3 units. General equation of circle is :- 𝑥−𝑎﷯﷮2﷯+ 𝑦−𝑏﷯﷮2﷯= 𝑟﷮2﷯ Given center is on y-axis & radius = 3 units ∴ Center = (0, b) Hence, our equation becomes 𝑥−0﷯﷮2﷯+ 𝑦−𝑏﷯﷮2﷯= 3﷯﷮2﷯ 𝑥﷮2﷯+ 𝑦−𝑏﷯﷮2﷯=9 Differentiating Both Sides w.r.t. 𝑥 2𝑥+2 𝑦−𝑏﷯ 𝑑𝑦﷮𝑑𝑥﷯−0﷯=0 2 𝑥+ 𝑦−𝑏﷯ 𝑑𝑦﷮𝑑𝑥﷯﷯=0 𝑥+ 𝑦−𝑏﷯ 𝑑𝑦﷮𝑑𝑥﷯=0 𝑦−𝑏﷯ 𝑑𝑦﷮𝑑𝑥﷯=−𝑥 𝑦−𝑐= −𝑥﷮ 𝑑𝑦﷮𝑑𝑥﷯﷯ Putting the value of 𝑦−𝑏 in equation (1) x2 + (y − c)2 = 9 𝑥﷮2﷯+ −𝑥﷮ 𝑑𝑦﷮𝑑𝑥﷯﷯﷯﷮2﷯=9 𝑥﷮2﷯+ 𝑥﷮2﷯﷮ 𝑑𝑦﷮𝑑𝑥﷯﷯﷮2﷯﷯=9 𝑥﷮2﷯ 𝑑𝑦﷮𝑑𝑥﷯﷯﷮2﷯+ 𝑥﷮2﷯﷮ 𝑑𝑦﷮𝑑𝑥﷯﷯﷮2﷯﷯=9 𝑥﷮2﷯ 𝑑𝑦﷮𝑑𝑥﷯﷯﷮2﷯+ 𝑥﷮2﷯=9 𝑑𝑦﷮𝑑𝑥﷯﷯﷮2﷯ 𝑥﷮2﷯ 𝑦﷮′﷯﷯﷮2﷯+ 𝑥﷮2﷯=9 𝑦﷮′﷯﷯﷮2﷯ 𝑥﷮2﷯ 𝑦﷮′﷯﷯﷮2﷯−9 𝑦﷮′﷯﷯﷮2﷯+ 𝑥﷮2﷯=0 𝑦﷮′﷯﷯﷮2﷯ 𝑥﷮2﷯−9﷯+ 𝑥﷮2﷯=0 ∴ 𝒙﷮𝟐﷯−𝟗﷯ 𝒚﷮′﷯﷯﷮𝟐﷯+ 𝒙﷮𝟐﷯=𝟎

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