# Example 7

Last updated at March 11, 2017 by Teachoo

Last updated at March 11, 2017 by Teachoo

Transcript

Example 7 Form the differential equation of the family of circles touching the 𝑥−𝑎𝑥𝑖𝑠 at origin. In Our Case , Center will be on 𝑦−𝑎𝑥𝑖𝑠 So, Center 0 , 𝑏 i.e. 𝑎=0 Also 𝑟𝑎𝑑𝑖𝑢𝑠=𝑏 So, Equation of Circle 𝑥−02+ 𝑦−𝑏2= 𝑏2 𝑥2+ 𝑦−𝑏2= 𝑏2 𝑥2+ 𝑦2−2𝑦𝑏+ 𝑏2= 𝑏2 𝑥2+ 𝑦2−2𝑦𝑏=0 Since there is one variable b, we differentiate once Diff. w.r.t. 𝑥 2𝑥+2𝑦 𝑑𝑦𝑑𝑥=2𝑏. 𝑑𝑦𝑑𝑥 𝑥+𝑦 𝑑𝑦𝑑𝑥=𝑏. 𝑑𝑦𝑑𝑥 𝑥 + 𝑦 𝑑𝑦𝑑𝑥 𝑑𝑦𝑑𝑥=𝑏 Putting Value of 𝑏 in (1) 𝑥2+ 𝑦2=2𝑦 𝑥 + 𝑦 𝑑𝑦𝑑𝑥 𝑑𝑦𝑑𝑥 𝑥2+ 𝑦2=2𝑦 𝑥 + 𝑦 𝑑𝑦𝑑𝑥 𝑑𝑦𝑑𝑥 𝑥2+ 𝑦2 𝑑𝑦𝑑𝑥=2𝑦 𝑥+𝑦 𝑑𝑦𝑑𝑥 𝑥2+ 𝑦2 𝑑𝑦𝑑𝑥=2𝑦 𝑥+2 𝑦2 𝑑𝑦𝑑𝑥 𝑥2 𝑑𝑦𝑑𝑥+ 𝑦2 𝑑𝑦𝑑𝑥=2𝑥𝑦+2 𝑦2 𝑑𝑦𝑑𝑥 𝑥2 𝑑𝑦𝑑𝑥+ 𝑦2 𝑑𝑦𝑑𝑥−2 𝑦2 𝑑𝑦𝑑𝑥=2𝑥𝑦 𝑥2 𝑑𝑦𝑑𝑥− 𝑦2 𝑑𝑦𝑑𝑥=2xy 𝑑𝑦𝑑𝑥 𝑥2− 𝑦2=2𝑥𝑦 𝒅𝒚𝒅𝒙= 𝟐𝒙𝒚 𝒙𝟐 − 𝒚𝟐

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About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.