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Example 7 - Family of circles touching x-axis at origin - Examples

Example 7 - Chapter 9 Class 12 Differential Equations - Part 2
Example 7 - Chapter 9 Class 12 Differential Equations - Part 3 Example 7 - Chapter 9 Class 12 Differential Equations - Part 4

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Example 7 Form the differential equation of the family of circles touching the x-axis at origin. We know that, Equation of Circle is (π‘₯βˆ’π‘Ž)^2+(π‘¦βˆ’π‘)^2=π‘Ÿ^2 Center =(π‘Ž,𝑏) Radius = π‘Ÿ Since the circle touches the x-axis at origin The center will be on the y-axis So, x-coordinate of center is 0 i.e. a = 0 ∴ Center = (0 , 𝑏) And, π‘Ÿπ‘Žπ‘‘π‘–π‘’π‘ =𝑏 So, Equation of Circle (π‘₯βˆ’0)^2+(π‘¦βˆ’π‘)^2=𝑏^2 π‘₯^2+(π‘¦βˆ’π‘)^2=𝑏^2 π‘₯^2+𝑦^2βˆ’2𝑦𝑏+𝑏^2=𝑏^2 π‘₯^2+𝑦^2βˆ’2𝑦𝑏=0 π‘₯^2+𝑦^2=2𝑦𝑏 Since there is one variable b, we differentiate once Diff. w.r.t. π‘₯ 2π‘₯+2𝑦 𝑑𝑦/𝑑π‘₯=2𝑏.𝑑𝑦/𝑑π‘₯ π‘₯+𝑦𝑦^β€²=𝑏.𝑦′ [(π‘₯ + 𝑦𝑦^β€²)/𝑦^β€² ]=𝑏 Putting Value of 𝑏 in (1) π‘₯^2+𝑦^2=2𝑦[(π‘₯ + 𝑦𝑦^β€²)/𝑦^β€² ] (π‘₯^2+𝑦^2 ) 𝑦^β€²=2𝑦(π‘₯+𝑦𝑦^β€² ) (π‘₯^2+𝑦^2 ) 𝑦^β€²=2𝑦π‘₯+2𝑦^2 𝑦^β€² π‘₯^2 𝑦^β€²+𝑦^2 𝑦^β€²=2𝑦π‘₯+2𝑦^2 𝑦^β€² π‘₯^2 𝑦^β€²+𝑦^2 𝑦^β€²βˆ’2𝑦^2 𝑦^β€²=2𝑦π‘₯ π‘₯^2 𝑦^β€²βˆ’π‘¦^2 𝑦^β€²=2𝑦π‘₯ 𝑦^β€² (π‘₯^2βˆ’π‘¦^2 )=2π‘₯𝑦 𝐲′=πŸπ’™π’š/(𝒙^𝟐 βˆ’ π’š^𝟐 )

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