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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

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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 is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.