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Example 8 - AOBA is part of ellipse 9x2 + y2 = 36, OA = 2

Example 8 - Chapter 8 Class 12 Application of Integrals - Part 2
Example 8 - Chapter 8 Class 12 Application of Integrals - Part 3 Example 8 - Chapter 8 Class 12 Application of Integrals - Part 4 Example 8 - Chapter 8 Class 12 Application of Integrals - Part 5 Example 8 - Chapter 8 Class 12 Application of Integrals - Part 6


Transcript

Example 8 In Fig AOBA is the part of the ellipse 9π‘₯2+𝑦2=36 in the first quadrant such that OA = 2 and OB = 6. Find the area between the arc AB and the chord AB Here, Area Area Required = Area ABC Area ABC Area ABC = Area OACB – Area βˆ† OAB Area OACB Area OACB = ∫_0^2▒𝑦𝑑π‘₯ Here, 𝑦 β†’ Equation of Ellpise 9π‘₯2+𝑦2=36 𝑦2=36βˆ’9π‘₯2 𝑦="Β±" √(36βˆ’9π‘₯^2 ) As OACB is in 1st quadrant, Value of 𝑦 will be positive ∴ 𝑦=√(36βˆ’9π‘₯^2 ) Area OACB = ∫_0^2▒𝑦𝑑π‘₯ = ∫_0^2β–’βˆš(36βˆ’9π‘₯^2 ) = ∫_0^2β–’βˆš(9(36/9βˆ’π‘₯^2)) = ∫_0^2β–’βˆš(9(4βˆ’π‘₯^2)) = ∫_0^2β–’βˆš(3^2 (2^2βˆ’π‘₯^2)) = 3∫_0^2β–’γ€–βˆš(2^2βˆ’π‘₯^2 ) 𝑑π‘₯ γ€— = 3[π‘₯/2 √(2^2βˆ’π‘₯^2 )+2^2/2 sin^(βˆ’1)⁑〖π‘₯/2γ€— ]_0^2 = 3 [2/2 √(2^2βˆ’2^2 )+2 sin^(βˆ’1)⁑〖2/2βˆ’0/2γ€— √(2^2βˆ’0^2 )βˆ’2 sin^(βˆ’1)⁑0 ] It is of form ∫1β–’γ€–βˆš(π‘Ž^2βˆ’π‘₯^2 ) 𝑑π‘₯=1/2 π‘₯√(π‘Ž^2βˆ’π‘₯^2 )γ€—+π‘Ž^2/2 〖𝑠𝑖𝑛〗^(βˆ’1)⁑〖π‘₯/π‘Ž+𝑐〗 Replacing a by 2 , we get = 3 [0+2 sin^(βˆ’1)⁑〖1βˆ’0βˆ’0γ€— ] = 3 [2 sin^(βˆ’1)⁑1 ] = 3 Γ— 2 Γ— πœ‹/2 = 3πœ‹ Area OAB Area OAB =∫_0^2▒𝑦𝑑π‘₯ Here, 𝑦 β†’ equation of line AB Equation of line between A(2, 0) & B(0, 6) is (𝑦 βˆ’ 0)/(π‘₯ βˆ’ 2)=(6 βˆ’ 0)/(0 βˆ’ 2) 𝑦/(π‘₯βˆ’2)=6/(βˆ’2) Eq. of line b/w (x1, y1) & (x2, y2) is (𝑦 βˆ’ 𝑦1)/(π‘₯ βˆ’ π‘₯1)=(𝑦2 βˆ’ 𝑦1)/(π‘₯2 βˆ’ π‘₯1) 𝑦/(π‘₯ βˆ’ 2)=βˆ’3 𝑦=βˆ’3(π‘₯βˆ’2) Area OBC =∫_0^2▒〖𝑦 𝑑π‘₯γ€— =∫_0^2β–’γ€–βˆ’3(π‘₯βˆ’2) 𝑑π‘₯γ€— = –3∫_0^2β–’γ€–(π‘₯βˆ’2) 𝑑π‘₯γ€— = –3 [π‘₯^2/2 βˆ’2π‘₯ ]_0^2 = – 3 [2^2/2βˆ’2 Γ—2βˆ’0^2/2βˆ’2 Γ—0] = –3 [βˆ’2] = 6 Area Required = Area OACB – Area βˆ† OAB =πŸ‘π…βˆ’πŸ” square units

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.