<s>[INST] The parabola $x^2=12y$ rolls without slipping around the parabola $x^2=-12y$ The parabola $x^2=12y$ rolls without slipping around the parabola $x^2=-12y$ then find the locus of focus of rolling parabola and also find the locus of vertex of rolling parabola 
 [/INST] 
\begin{align}
\text{Static parabola: }f(x)&=-\tfrac1{12}x^2
,\\
f'(x)&=-\tfrac1{6}x
.
\end{align}
Tangent line at $x=x_t$: 
\begin{align}
f_t(x,x_t)&=
f'(x_t)\,(x-x_t)+f(x_t)
\\
&=
-\tfrac1{6}x_t\,(x-x_t)-\tfrac1{12}x_t^2
\\
&=
-\tfrac16\,x_t\,x+\tfrac1{12}\,x_t^2
.
\end{align}  
\begin{align}
|OD|&=\tfrac12x_t \quad\text{(why?)}
,\\
\tan\theta&=\tfrac16\,x_t
,\\
\sin\theta&=
\frac{\tan\theta}{\sqrt{1+\tan^2\theta}}
=\frac{x_t}{\sqrt{x_t^2+36}}
,\\
\cos\theta&=
\frac{6}{\sqrt{x_t^2+36}}
,\\
|OO'|&=2|OD|\sin\theta
=\frac{x_t^2}{\sqrt{x_t^2+36}}
,\\
O'_x&=|OO'|\,\cos(90^\circ-\theta)=|OB|\,\sin\theta
=\frac{x_t^3}{x_t^2+36}
,\\
O'_y&=|OO'|\,\sin(90^\circ-\theta)=|OB|\,\cos\theta
=
\frac{6x_t^2}{x_t^2+36}
.\\
\end{align}
Hence, the locus of vertex of rolling parabola 
is defined parametrically as
\begin{align}
O'(x_t)&=
\left(\frac{t^3}{t^2+36}, \frac{6t^2}{t^2+36}\right)
,\quad t\in\mathbb{R}
.
\end{align}
This can be used as a base to derive the locus of the rolling focus.
</s>