If `sqrt(1-x^2) + sqrt(1-y^2) = a(x-y), show dy/dx = sqrt((1-y^2)/(1-x^2))`

Explanation

`sqrt(1-x^(2))+sqrt(1-y^(2))=a(x-y)" … (1)"`
Put `x=sintheta, y=sinphi`
`therefore theta =sin^(-1)x,phi=sin^(-1)y`
`therefore` (1) becomes, `sqrt(1-sin^(2)theta)+sqrt(1-sin^(2)phi)=a(sintheta-sinphi)`
`therefore cos theta+cosphi=a(sintheta-sinphi)`
`therefore 2cos((theta+phi)/(2)).cos((theta-phi)/(2))=axx2cos((theta+phi)/(2)).sin.((theta-phi)/(2))`
`thereforecos((theta-phi)/(2))=asin((theta-phi)/(2))`
`therefore (cos((theta-phi)/(2)))/(sin((theta-phi)/(2)))=a`
`therefore cot((theta-phi)/(2))=a`
`therefore(theta-phi)/(2)=cot^(-1)a`
`therefore theta-phi=2cot^(-1)a`
`therefore sin^(-1)x-sin^(-1)y=2cot^(-1)a`
Differentiating both sides w.r.t. x, we get,
`(1)/(sqrt(1-x^(2)))-(1)/(sqrt(1-y^(2))).(dy)/(dx)=0`
`therefore (1)/(sqrt(1-y^(2)))(dy)/(dx)=(1)/(sqrt(1-x^(2)))`
`therefore (dy)/(dx)=sqrt((1-y^(2))/(1-x^(2)))`.