The curve $C$ has equation $y=\frac{2x-a}{2{{x}^{2}}-b}$, where $a$ and $b$ are positive integers. $

Question

The curve $C$ has equation $y=\frac{2x-a}{2{{x}^{2}}-b}$, where $a$ and $b$ are positive integers. $C$ passes through point $\left( \frac{9}{2},0 \right)$ and the equation of an asymptote of $C$ is $x=3$. Show that $a=9$ and $b=18$. [2] Sketch $C$, stating clearly the equations of asymptotes, the $x-$coordinates of turning points and axial intercepts. [4] Hence, giving your reasons, deduce the range of values of $h$ such that the graph of $\frac{{{\left( x-8 \right)}^{2}}}{{{h}^{2}}}+\frac{{{y}^{2}}}{100}=1$, where $h$ is a positive integer, intersects $C$ at exactly $6$ distinct points. [3]

Accepted Answer

(i) $a = 9$, $b = 18$; (ii) VA: $x = \pm 3$, HA: $y = 0$; (iii) $h \geq 12$

SAJC Graphing Techniques Tutorial Q1

SAJC Graphing Techniques Tutorial Q1

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