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If sqrt(1-x^2) + sqrt(1-y^2) = a(x-y), s...

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

Text Solution

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We have,
`sqrt(1-x^(2))+sqrt(1-y^(2))=a(x-y)`
On putting `x = sin alpha` and `y = sin beta`, we get
`sqrt(1-sin^(2)alpha)+sqrt(1-sin^(2)beta)=a(sinalpha-sinbeta)`
`rArr cosalpha+cosbeta=a(sinalpha-sinbeta)`
`rArr 2cos'(alpha+beta)/(2).cos'(alpha-beta)/(2)=a(2cos'(alpha+beta)/(2).sin'(alpha-beta)/(2))`
`rArr cos'(alpha-beta)/(2)=asin'(alpha-beta)/(2)`
`rArr cot(alpha-beta)/(2)=a`
`rArr (alpha-beta)/(2)=cot^(-1)a`
`rArr alpha - beta = 2cot^(-1)a`
`rArr sin^(-1)x -sin^(-1)y= 2cot^(-1)a`, `[:'x = sinalpha"and"y=sinbeta]`
On differentiating both sides w.r.t.x, we get
`(1)/(sqrt(1-x^(2)))-(1)/(sqrt(1-y^(2)))(dy)/(dx) = 0`
` :. (dy)/(dx) = (sqrt(1-y^(2)))/(sqrt(1-x^(2)))= sqrt((1-y^(2))/(1-x^(2)))`
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