If yβ1-x ^{ 2 } +xβ1-y ^{ 2 } =1, then prove that dy/dx=-β1 - y ^{ 2 } /1 - x ^{ 2 }
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CBSE Class 12 Sample Paper for 2023 Boards
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CBSE Class 12 Sample Paper for 2023 Boards
Last updated at March 22, 2023 by Teachoo
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Question 24 If π¦β(1βπ₯^2 )+π₯β(1βπ¦^2 )=1, then prove that dy/dx=ββ((1 β π¦^2)/(1 β π₯^2 )) Finding π π/π π would be complicated here To make life easy, we substitute x = sin A y = sin B (As β(1βπ₯^2 )= β(1βsin^2β‘π΄ )=β(cos^2β‘π΄ )) And then solve Letβs substitute x = sin A y = sin B in our equation Now π¦β(1βπ₯^2 ) + π₯β(1βπ¦^2 ) = 1 Putting x = sin A and y = sin B π¬π’π§ πβ(πβγπ¬π’π§γ^πβ‘π¨ ) + π¬π’π§ πβ(πβγπππγ^πβ‘π© ) = 1 sin Bβ(cos^2β‘π΄ ) + sin Aβ(cos^2β‘π΅ ) = a (sin A β 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 /π A + B = π /π Putting back values of A and B sin^(β1)β‘π₯+sin^(β1)β‘π¦=π/2 Differentiating w.r.t x 1/β(1 β π₯^2 )β1/β(1 β π¦^2 )Γππ¦/ππ₯=0 1/β(1 β π₯^2 )=1/β(1 β π¦^2 )Γππ¦/ππ₯ β(1 β π¦^2 )/β(1 β π₯^2 )=ππ¦/ππ₯ π π/(π π ) = β(π β π^π )/β(π β π^π )