If `y sqrt(1-x^2)+x sqrt(1-y^2)=1`. Prove that `dy/dx=-sqrt((1-y^2)/(1-x^2))`

If `y sqrt(1-x^2)+x sqrt(1-y^2)=1`. Prove that `dy/dx=-sqrt((1-y^2)/(1

1.	If `y sqrt(1-x^2)+x sqrt(1-y^2)=1`. Prove that `dy/dx=-sqrt((1-y^2)/(1-x^2))`
Answer» We have `xsqrt(1-y^(2))+ysqrt(1-x^(2))=1." …(i)"` Putting `x = sin theta and y = cos phi` in (i), we get `sin theta cos phi+cos theta sin phi=1` `rArr sin(theta+phi)=1` `rArr (theta+phi)=sin^(-1)(1)` `rArr sin^(-1)x+sin^(-1)y=(pi)/(2)." ...(ii)"` On differentiating both sides of (ii) w.r.t. x, we get `(1)/(sqrt(1-x^(2)))+(1)/(sqrt(1-y^(2))).(dy)/(dx)=0.` `therefore (dy)/(dx)=-sqrt((1-y^(2))/(1-x^(2))).`

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