Question

If `cos((x^2 - y^2)/(x^2 + y^2))` = log a, show that `dy/dx = y/x`

Answer 1

`cos((x^2 - y^2)/(x^2 + y^2))` = log a

∴ `(x^2 - y^2)/(x^2 + y^2)` = cos^–1(log a)

∴ Let `(x^2 - y^2)/(x^2 + y^2)` = k   (constant)

∴ kx² + ky² = x² – y²

∴ y²(k + 1) = x²(1 – k)

∴ `y^2/x^2 = (1 - k)/(k + 1)`

∴ `y/x = sqrt((1 - k)/(1 + k)`

Differentiating w.r.t.x,

`(x dy/dx - y)/x^2` = 0

∴ `x dy/dx - y` = 0

∴ `x dy/dx` = y

∴ `dy/dx = y/x`

Hence proved.