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Ex 9.3, 8 - Family of ellipses having foci on y-axis, center

Ex 9.3, 8 - Chapter 9 Class 12 Differential Equations - Part 2
Ex 9.3, 8 - Chapter 9 Class 12 Differential Equations - Part 3

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Question 8 Form the differential equation of the family of ellipses having foci on π‘¦βˆ’π‘Žπ‘₯𝑖𝑠 and center at origin. Equation of ellipse having center at origin (0, 0) & foci on y-axis is π‘₯^2/𝑏^2 +𝑦^2/π‘Ž^2 =1 ∴ Differentiating Both Sides w.r.t. π‘₯ 𝑑/𝑑π‘₯ [π‘₯^2/𝑏^2 +𝑦^2/π‘Ž^2 ] = (𝑑(1))/𝑑π‘₯ 1/𝑏^2 [2π‘₯]+1/π‘Ž^2 [2𝑦] 𝑑𝑦/𝑑π‘₯=0 2π‘₯/𝑏^2 +2𝑦/π‘Ž^2 . 𝑑𝑦/𝑑π‘₯=0 Since it has two variables, we will differentiate twice 2𝑦/π‘Ž^2 𝑦′=(βˆ’2π‘₯)/𝑏^2 𝑦/π‘Ž^2 𝑦′=(βˆ’π‘₯)/𝑏^2 (𝑦/π‘₯)𝑦′=(βˆ’π‘Ž^2)/γ€– 𝑏〗^2 (𝑦𝑦^β€²)/π‘₯ = (βˆ’π‘Ž^2)/𝑏^2 Again differentiating both sides w.r.t. x ((𝑦𝑦^β€² )^β€² π‘₯ βˆ’ (𝑑π‘₯/𝑑π‘₯)(𝑦𝑦^β€² ))/π‘₯^2 =0 (𝑦𝑦^β€² )^β€² π‘₯ βˆ’ (1)(𝑦𝑦^β€² )=πŸŽΓ—π’™^𝟐 (𝑦𝑦^β€² )^β€² π‘₯ βˆ’π‘¦π‘¦^β€²=𝟎 (Using Quotient rule and Diff. of constant is 0) (π’šπ’š^β€² )^β€² π‘₯ βˆ’π‘¦π‘¦^β€²=0 (π’š^β€² π’š^β€²+π’šπ’šβ€²β€²)π‘₯ βˆ’π‘¦π‘¦^β€²=0 (〖𝑦^β€²γ€—^2+𝑦𝑦′′)π‘₯ βˆ’π‘¦π‘¦^β€²=0 π‘₯〖𝑦^β€²γ€—^2+π‘₯𝑦𝑦^β€²β€²βˆ’π‘¦π‘¦^β€²=0 π’™π’šπ’š^β€²β€²+π’™γ€–π’š^β€²γ€—^πŸβˆ’π’šπ’š^β€²=𝟎 (Using Product rule)

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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.