Question

Find $\frac{dy}{dx}$, if x and y are connected parametrically by the equations given in questions without eliminating the parameter.

$x=a(cost+logtan\frac{t}{2}),y=asint.$

Answer 1

Given, $x=a(cost+logtan\frac{t}{2})$

Differentiating w.r.t. t, we get

$\frac{dx}{dt}=a\{-sint+\frac{1}{tan\frac{t}{2}}.se{c}^2\frac{t}{2}.\frac{1}{2}\}[\because \frac{d}{dx}(log|x|)=\frac{1}{x}]=a\{-sint+\frac{1}{2sin\frac{t}{2}cos\frac{t}{2}}\}=a\{-sint+\frac{1}{sint}\}=a\left\{\frac{1-si{n}^{2}t}{sint}\right\}(\because sint=2sin\frac{t}{2}cos\frac{t}{2})\Rightarrow \frac{dx}{dt}=\frac{aco{s}^{2}t}{sint}(\because 1-si{n}^{2}t=co{s}^{2}t)........\left(i\right)Also,y=asint\Rightarrow \frac{dy}{dt}=acost.....\left(ii\right)FromEqs.\left(i\right)and\left(ii\right),\therefore \frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}\Rightarrow \frac{dy}{dx}=\frac{acost}{\frac{aco{s}^{2}t}{sint}}=tant$