Question

If \(x\) and \(y\) are connected parametrically by the equation,without eliminating the parameter,find \(\frac{dy}{dx}\).
\(x=a(cos\,t+log\,tan\frac{t}{2}),y=a\,sint\)

Answer 1

The correct answer is \(tan\,t\)
The given equations are \(x=a(cos\,t+log\,tan\frac{t}{2}),y=a\,sint\)
Then,\(\frac{dx}{dt}=a[\frac{d}{dt}(cost)+\frac{d}{dt}(log\,tan\frac{t}{2})]\)
\(=a[-sin\,t+\frac{1}{tan\frac{t}{2}}.\frac{d}{dt}(tan\frac{t}{2})]\)
\(=a[-sin\,t+cot\frac{t}{2}.sec^2\frac{t}{2}.\frac{d}{dt}(\frac{t}{2})]\)
\(=a[-sint+\frac{cos\frac{t}{2}}{sin\frac{t}{2}}\times \frac{1}{cos^2\frac{t}{2}}.\frac{1}{2}]\)
\(=a[-sint+\frac{1}{2sin\frac{t}{2}cos\frac{t}{2}}]\)
\(=a(-sint+\frac{1}{sint})\)
\(=a(\frac{-sin^2t+1}{sint})\)
\(=a\frac{cos^2t}{sint}\)
\(\frac{dy}{dt}=a\frac{d}{dt}(sint)=acost\)
\(∴\frac{dy}{dx}=\frac{(\frac{dy}{dt})}{(\frac{dx}{dt})}=\frac{acost}{(\frac{acos^2t}{sint})}\)
\(=\frac{sint}{cost}=tan\,t\)