The curve $C$ has equation \[y = x - 2 + \frac{3k(x - 2)}{x^2 - 2k}, \quad \text{where}\hspace{0.5

Question

The curve $C$ has equation $$y = x - 2 + \frac{3k(x - 2)}{x^2 - 2k}, \quad \text{where}\hspace{0.5em} k > 0, \ k \ne 2.$$ Without the use of a calculator, find the coordinates of any points of intersection of $C$ with the axes.[3] Write down the equations of the asymptotes of $C$ in terms of $k$.[2] It is now given that $k = 1$. Show that the $x$-coordinates of the turning points of $C$ satisfy the equation $$x^4 - 7x^2 + 12x - 2 = 0.$$ Hence find these $x$-coordinates.[3] Sketch $C$, labelling all essential features.[3]

Accepted Answer

(a) $(0,1)$ and $(2,0)$ (b) $x=\pm\sqrt{2k}$ and $y=x-2$ (c) $x=-3.29$ or $x=0.187$, to 3 significant figures (d) See sketch.

2024 EJC Promo Q12

2024 EJC Promo Q12

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