If y√1-x 2 +x√1-y 2 =1, then prove that dy/dx=-√1 - y 2 /1 - x 2

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Question 24 If π‘¦βˆš(1βˆ’π‘₯^2 )+π‘₯√(1βˆ’π‘¦^2 )=1, then prove that dy/dx=βˆ’βˆš((1 βˆ’ 𝑦^2)/(1 βˆ’ π‘₯^2 )) Finding π’…π’š/𝒅𝒙 would be complicated here To make life easy, we substitute x = sin A y = sin B (As √(1βˆ’π‘₯^2 )= √(1βˆ’sin^2⁑𝐴 )=√(cos^2⁑𝐴 )) And then solve Let’s substitute x = sin A y = sin B in our equation Now π‘¦βˆš(1βˆ’π‘₯^2 ) + π‘₯√(1βˆ’π‘¦^2 ) = 1 Putting x = sin A and y = sin B 𝐬𝐒𝐧 𝐁√(πŸβˆ’γ€–π¬π’π§γ€—^πŸβ‘π‘¨ ) + 𝐬𝐒𝐧 π€βˆš(πŸβˆ’γ€–π’”π’Šπ’γ€—^πŸβ‘π‘© ) = 1 sin B√(cos^2⁑𝐴 ) + sin A√(cos^2⁑𝐡 ) = a (sin A βˆ’ sin B) sin B cos A + sin A cos B = 1 sin A cos B + sin B cos A = 1 sin (A + B) = 1 sin (A + B) = sin 𝝅/𝟐 A + B = 𝝅/𝟐 Putting back values of A and B sin^(βˆ’1)⁑π‘₯+sin^(βˆ’1)⁑𝑦=πœ‹/2 Differentiating w.r.t x 1/√(1 βˆ’ π‘₯^2 )βˆ’1/√(1 βˆ’ 𝑦^2 )×𝑑𝑦/𝑑π‘₯=0 1/√(1 βˆ’ π‘₯^2 )=1/√(1 βˆ’ 𝑦^2 )×𝑑𝑦/𝑑π‘₯ √(1 βˆ’ 𝑦^2 )/√(1 βˆ’ π‘₯^2 )=𝑑𝑦/𝑑π‘₯ π’…π’š/(𝒅𝒙 ) = √(𝟏 βˆ’ π’š^𝟐 )/√(𝟏 βˆ’ 𝒙^𝟐 )

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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.