Question

If x and y are connected parametrically by the equation, without eliminating the parameter, find dy/dx x=a(θ -sinθ) y=a(1+cosθ )

Answer 1

Given, x = a(θ - sinθ) , y = a(1 + cosθ)

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

y = a(1 + cosθ)
now, differentiate y with respect to θ,
dy/dθ = d{a(1 + cosθ)}/dθ
= a{d1/dθ + d(cosθ)/dθ}
= a{0 - sinθ} = - asinθ -----(2)

dividing equations (2) by (1),
{dy/dθ}/{dx/dθ} = (-asinθ)/a(1-cosθ)
dy/dx = sinθ/(cosθ -1)