Question

If x and y are connected parametrically by the equations, without eliminating the parameter, find `bb(dy/dx)`.

x = a (cos θ + θ sin θ), y = a (sin θ – θ cos θ)

Answer 1

Here x = a (cos θ + θ sin θ) ...(1)

y = y = a (sin θ – θ cos θ) ...(2)

Differentiating (1) and (2) w.r.t. θ, we get

`dx/(d θ)` = a [−sin θ + θ × cos θ + sin θ]

= a  θ cos θ

`dy/(d theta)` = a [cos θ − (θ (−sin θ) + cos θ)]

= a [cos θ + θ sin θ − cos θ]

= a θ sin θ

`dy/dx = (dy/(dθ))/(dx/(dθ))`

= `(a θ sin θ)/(a θ cos θ)`

= tan θ