Question

If x and y are connected parametrically by the equation, without eliminating the parameter, find.dy/dx x=a(cosθ +θ sinθ ), y=(sinθ -θ cosθ )

Answer 1

x = a(cosθ + θsinθ) and y = a(sinθ - θcosθ)

x = a(cosθ + θsinθ)
now differentiate x with respect to θ,
dx/dθ = a{d(cosθ)/dθ + θ.d(sinθ)/dθ + sinθ.dθ/dθ}
= a(-sinθ + θ.cosθ + sinθ)
= aθ.cosθ -------(1)

y = a(sinθ - θcosθ)
now differentiate y with respect to θ,
dy/dθ = a{d(sinθ)/dθ - θ.d(cosθ)/dθ - cosθ.dθ/dθ}
= a(cosθ + θ.sinθ - cosθ)
= aθ.sinθ ------(2)

dividing equations (2) by (1),
{dy/dθ}/{dx/dθ} = (aθ.sinθ)/(aθ.cosθ)
dy/dx = tanθ

hence, dy/dx = tanθ