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Ex 9.3, 6 - Form differential equation of family of circles

Ex 9.3, 6 - Chapter 9 Class 12 Differential Equations - Part 2
Ex 9.3, 6 - Chapter 9 Class 12 Differential Equations - Part 3


Transcript

Ex 9.3, 6 Form the differential equation of the family of circle touching the ğ‘¦âˆ’ğ‘Žğ‘¥ğ‘–ğ‘  at origin. General Equation of Circle (ğ‘¥âˆ’ğ‘Ž)^2+(𝑦−𝑏)^2=𝑟^2 where Centre at (ğ‘Ž , 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 (ğ‘¥âˆ’ğ‘Ž)^2+(𝑦−0)^2=ğ‘Ž^2 (ğ‘¥âˆ’ğ‘Ž)^2+𝑦^2=ğ‘Ž^2 𝑥^2+ğ‘Ž^2−2ğ‘Žğ‘¥+𝑦^2=ğ‘Ž^2 𝑥^2−2ğ‘Žğ‘¥+𝑦^2=ğ‘Ž^2âˆ’ğ‘Ž^2 𝑥^2−2ğ‘Žğ‘¥+𝑦^2=0 2ğ‘Žğ‘¥=𝑥^2+𝑦^2 Differentiating Both Sides w.r.t. 𝑥 (𝑑(2ğ‘Žğ‘¥))/𝑑𝑥=𝑑(𝑥^2 )/𝑑𝑥+𝑑(𝑦^2 )/𝑑𝑥 2a = 2x + 2y 𝑑𝑦/𝑑𝑥 a = x + yy’ …(1) …(2) From (1) 2ğ‘Žğ‘¥=𝑥^2+𝑦^2 Putting value of a from (2) 2𝑥(𝑥+𝑦𝑦^′)=𝑥^2+𝑦^2 2𝑥^2+2𝑥𝑦𝑦^′=𝑥^2+𝑦^2 2𝑥^2−𝑥^2+2𝑥𝑦𝑦^′=+𝑦^2 𝟐𝒙𝒚𝒚^′+𝒙^𝟐=𝒚^𝟐 is the required differential equation.

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.