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Misc 9 - Find area of smaller region x2/a2 + y2/b2 = 1 - Area between curve and line

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Misc 9 Find the area of the smaller region bounded by the ellipse 𝑥﷮2﷯﷮ 𝑎﷮2﷯﷯+ 𝑦﷮2﷯﷮ 𝑏﷮2﷯﷯=1 & 𝑥﷮𝑎﷯ + 𝑦﷮𝑏﷯ = 1 Step 1: Drawing figure 𝒙﷮𝟐﷯﷮ 𝒂﷮𝟐﷯﷯+ 𝒚﷮𝟐﷯﷮ 𝒃﷮𝟐﷯﷯=𝟏 is an which is a equation ellipse with 𝑥−𝑎𝑥𝑖𝑠 as principle 𝑎𝑥𝑖𝑠 For 𝒙﷮𝒂﷯ + 𝒚﷮𝒃﷯ = 1 Points A(a, 0) and B(0, b) passes through both line and ellipse Required Area Required Area = Area OACB – Area OAB Area OACB Area OACB = 0﷮𝑎﷮𝑦 𝑑𝑥﷯ 𝑦 → Equation of ellipse 𝑥﷮2﷯﷮ 𝑎﷮2﷯﷯+ 𝑦﷮2﷯﷮ 𝑏﷮2﷯﷯=1 𝑦﷮2﷯﷮ 𝑏﷮2﷯﷯=1− 𝑥﷮2﷯﷮ 𝑎﷮2﷯﷯ 𝑦﷮2﷯= 𝑏﷮2﷯ 1− 𝑥﷮2﷯﷮ 𝑎﷮2﷯﷯﷯ 𝑦= ﷮ 𝑏﷮2﷯ 1− 𝑥﷮2﷯﷮ 𝑎﷮2﷯﷯﷯﷯ 𝑦=𝑏 ﷮1− 𝑥﷮2﷯﷮ 𝑎﷮2﷯﷯﷯ Therefore, Area OACB = 0﷮𝑎﷮𝑏 ﷮1− 𝑥﷮2﷯﷮ 𝑎﷮2﷯﷯﷯﷯𝑑𝑥 =b 0﷮𝑎﷮ ﷮ 𝑎﷮2﷯ − 𝑥﷮2﷯﷮ 𝑎﷮2﷯﷯ ﷯ 𝑑𝑥 ﷯ = 𝑏﷮𝑎﷯ 0﷮𝑎﷮ ﷮ 𝑎﷮2﷯− 𝑥﷮2﷯﷯ 𝑑𝑥 ﷯ = 𝑏﷮𝑎﷯ 1﷮2﷯𝑥 ﷮ 𝑎﷮2﷯− 𝑥﷮2﷯﷯+ 𝑎﷮2﷯﷮2﷯ sin﷮−1﷯﷮ 𝑥﷮𝑎﷯﷯﷯﷮0﷮𝑎﷯ = 𝑏﷮𝑎﷯ 1﷮2﷯.𝑎 ﷮ 𝑎﷮2﷯− 𝑎﷮2﷯﷯+ 𝑎﷮2﷯﷮2﷯ sin﷮−1﷯﷮ 𝑎﷮𝑎﷯− 1﷮2﷯ 0 ﷮ 𝑎﷮2﷯− 0﷮2﷯﷯+ 𝑎﷮2﷯﷮2﷯ sin﷮−1﷯﷮ 0﷮𝑎﷯﷯﷯﷯﷯ = 𝑏﷮𝑎﷯ 0+ 𝑎﷮2﷯﷮2﷯. 𝒔𝒊𝒏﷮−𝟏﷯﷮𝟏﷯−0−0﷯ = 𝑏﷮𝑎﷯ 0+ 𝑎﷮2﷯﷮2﷯. 𝝅﷮𝟐﷯ ﷯ = 𝑏﷮𝑎﷯ 𝑎﷮2﷯﷮2﷯ 𝜋﷮2﷯ = 𝜋 𝑎𝑏 ﷮4﷯ Area OAB Area OAB = 0﷮𝑎﷮𝑦 𝑑𝑥﷯ 𝑦 → Equation of line 𝑥﷮𝑎﷯+ 𝑦﷮𝑏﷯=1 𝑦﷮𝑏﷯=1− 𝑥﷮𝑎﷯ 𝑦=𝑏 1− 𝑥﷮𝑎﷯﷯ Therefore, Area OAB = 0﷮𝑎﷮𝑏 1− 𝑥﷮𝑎﷯﷯𝑑𝑥﷯ = 𝑏 𝑥− 𝑥﷮2﷯﷮2𝑎﷯﷯﷮0﷮𝑎﷯ = 𝑏 𝑎− 𝑎﷮2﷯﷮2𝑎﷯− 0− 0﷮2﷯﷮2𝑎﷯﷯﷯ = 𝑏 𝑎− 𝑎﷮2﷯−0﷯ = 𝑎𝑏﷮2﷯ ∴ Area Required = Area OACB – Area OAB = 𝜋 𝑎𝑏 ﷮4﷯− 𝑎𝑏﷮2﷯ = 𝑎𝑏﷮2﷯ 𝜋﷮2﷯−1﷯ = 𝑎𝑏﷮2﷯ 𝜋−2﷮2﷯﷯ = 𝒂𝒃﷮𝟒﷯ 𝝅−𝟐﷯ square units

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CA Maninder Singh is a Chartered Accountant for the past 8 years. He provides courses for Practical Accounts, Taxation and Efiling at teachoo.com .
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