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

  1. Chapter 9 Class 12 Differential Equations (Term 2)
  2. Serial order wise


Ex 9.3, 6 Form the differential equation of the family of circle touching the ๐‘ฆโˆ’๐‘Ž๐‘ฅ๐‘–๐‘  at origin. General Equation of Circle (๐‘ฅโˆ’๐‘Ž)^2+(๐‘ฆโˆ’๐‘)^2=๐‘Ÿ^2 where Centre at (๐‘Ž , ๐‘) 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 (๐‘ฅโˆ’๐‘Ž)^2+(๐‘ฆโˆ’0)^2=๐‘Ž^2 (๐‘ฅโˆ’๐‘Ž)^2+๐‘ฆ^2=๐‘Ž^2 ๐‘ฅ^2+๐‘Ž^2โˆ’2๐‘Ž๐‘ฅ+๐‘ฆ^2=๐‘Ž^2 ๐‘ฅ^2โˆ’2๐‘Ž๐‘ฅ+๐‘ฆ^2=๐‘Ž^2โˆ’๐‘Ž^2 ๐‘ฅ^2โˆ’2๐‘Ž๐‘ฅ+๐‘ฆ^2=0 2๐‘Ž๐‘ฅ=๐‘ฅ^2+๐‘ฆ^2 Differentiating Both Sides w.r.t. ๐‘ฅ (๐‘‘(2๐‘Ž๐‘ฅ))/๐‘‘๐‘ฅ=๐‘‘(๐‘ฅ^2 )/๐‘‘๐‘ฅ+๐‘‘(๐‘ฆ^2 )/๐‘‘๐‘ฅ 2a = 2x + 2y ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ a = x + yyโ€™ โ€ฆ(1) โ€ฆ(2) From (1) 2๐‘Ž๐‘ฅ=๐‘ฅ^2+๐‘ฆ^2 Putting value of a from (2) 2๐‘ฅ(๐‘ฅ+๐‘ฆ๐‘ฆ^โ€ฒ)=๐‘ฅ^2+๐‘ฆ^2 2๐‘ฅ^2+2๐‘ฅ๐‘ฆ๐‘ฆ^โ€ฒ=๐‘ฅ^2+๐‘ฆ^2 2๐‘ฅ^2โˆ’๐‘ฅ^2+2๐‘ฅ๐‘ฆ๐‘ฆ^โ€ฒ=+๐‘ฆ^2 ๐Ÿ๐’™๐’š๐’š^โ€ฒ+๐’™^๐Ÿ=๐’š^๐Ÿ 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 10 years. He provides courses for Maths and Science at Teachoo.