Question

If \(x\) and \(y\) are connected parametrically by the equation,without eliminating the parameter,find \(\frac{dy}{dx}.\)
\(x=2at^2,y=at^4\)

Answer 1

The correct answer is \(∴\frac{dy}{dx}=\frac{(\frac{dy}{dt})}{(\frac{dx}{dt})}=\frac{4at^3}{4at}=t^2\)
The given equations are \(x=2at^2,y=at^4\)
Then,\(\frac{dx}{dt}=\frac{d}{dt}(2at^2)=2a.\frac{d}{dt}(t^2)=2a.2t=4at\)
\(\frac{dy}{dt}=\frac{d}{dt}(at^4)=a.\frac{d}{dt}(t^4)=a.4.t^3=4at^3\)
\(∴\frac{dy}{dx}=\frac{(\frac{dy}{dt})}{(\frac{dx}{dt})}=\frac{4at^3}{4at}=t^2\)