Question

If \(x=at^2,\ y=2at\), find the value of \(\dfrac{dy}{dx}\).

Hint:

Differentiate both x and y with respect to t, then divide.

Answer 1

Step 1: Eliminating the parameter first:
From \(y=2at\), \(t=\dfrac{y}{2a}\); substituting into \(x=at^2\) gives \(x=a\left(\dfrac{y}{2a}\right)^2=\dfrac{y^2}{4a}\).

Step 2: Differentiating implicitly:
\(x=\dfrac{y^2}{4a}\Rightarrow \dfrac{dx}{dy}=\dfrac{2y}{4a}=\dfrac{y}{2a}=\dfrac{2at}{2a}=t\).

Step 3: Inverting:
\(\dfrac{dy}{dx}=\dfrac{1}{dx/dy}=\dfrac1t\).

Final Answer:
\[ \boxed{\dfrac{dy}{dx}=\dfrac1t} \]