1.

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

Answer» `therefore sqrt(1-x^2)+sqrt(1-y^2)=a (x-y)`
`rArr sqrt(1-sin^2A)+sqrt(1-sin^2 )=B=a (sinA-sin B)`
`rArr cosA +cos B = a(sin A-sin B)`
`rArr 2cos""(A+B)/2.cos"" (A-B)/2=a 2 cos ""(A+B)/2 sin""(A-B)/2`
`rArr cos""(A-B)/2=a.sin""(A-B)/2`
`rArr cos""(A-B)/2=cot^-1a`
`rArr (A-B)/2=cot^-1a`
`rArr A-B = 2 cot^-1a`
`rArr sin^-1x sin ^1y=2 cot ^-1 a`
Differentiate both sides with respect to x `1/sqrt(1-x)-1/sqrt(1-y)dy/dx=0`
`rArr dy/dx=(sqrt(1-y^2))/(sqrt(1-x^2))`


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