Review question

# Can we show these two cubic curves touch? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8543

## Question

A curve is given by the equation $\begin{equation*} y = ax^3 - 6ax^2 + (12a + 12)x - (8a + 16), \label{eq:1}\tag{*} \end{equation*}$ where $a$ is a real number. Show that this curve touches the curve with equation $\begin{equation*} y = x^3 \label{eq:2}\tag{*{*}} \end{equation*}$

at $(2,8)$. Determine the coordinates of any other point of intersection of the two curves.

1. Sketch on the same axes the curves $\eqref{eq:1}$ and $\eqref{eq:2}$ when $a = 2$.

2. Sketch on the same axes the curves $\eqref{eq:1}$ and $\eqref{eq:2}$ when $a = 1$.

3. Sketch on the same axes the curves $\eqref{eq:1}$ and $\eqref{eq:2}$ when $a = -2$.