IAL 2022 Jan Q3 | 國際數學站

Figure 1

Figure 1 shows a sketch of the curve $C_1$ with equation

\begin{align*} y =\,& \frac{4x}{4 - |x|}\\[2mm] \end{align*}

and the curve $C_2$ with equation

\begin{align*} y =\,& x^2 - 8x\\[2mm] \end{align*}

For $x > 0$ , $C_1$ has equation $y = \frac{4x}{4 - x}$

(a) Use algebra to show that $C_1$ touches $C_2$ at a point $P$ , stating the coordinates of $P$ .

(5)

(b) Hence or otherwise, using algebra, solve the inequality

\begin{align*} x^2 - 8x >\,& \frac{4x}{4 - |x|}\\[2mm] \end{align*}

(6)