Check sibling questions

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

Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Transcript

Question 4 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 )

Ask a doubt
Davneet Singh's photo - Co-founder, Teachoo

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.