Question

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

Answer 1

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