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Question
If `sqrt(1-x^2) + sqrt(1- y^2)` = a(x − y), show that dy/dx = `sqrt((1-y^2)/(1-x^2))`
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Solution
Given, `sqrt(1 - x^2) + sqrt(1 - y^2) = a(x - y)`
Put x = sin θ and y = sin Φ
θ = sin−1 x and Φ = sin−1 y
Now, `sqrt(1 - sin^2 θ) + sqrt(1 − sin^2 "Φ") = a (sin θ − sin "Φ")`
cos θ + cos Φ = a(sin θ − sin Φ)
2 cos `((θ + "Φ")/2) cos ((θ − "Φ")/2) = 2a cos ((θ + "Φ")/2). sin ((θ − "Φ")/2)`
`therefore cos((θ − "Φ")/2)/sin ((θ − "Φ")/2) = a`
`therefore cot ((θ − "Φ")/2) = a`
`therefore (θ − "Φ")/2 = cot^-1 a`
∴ θ − Φ = 2 cot−1 a
∴ sin−1 x − sin−1 y = 2 cot−1 a
Differentiating 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)/(1 - x^2))`
Hence Proved
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