The curve $C$ has equation $y=\frac{{{x}^{2}}+ax+b}{c-x}$. The vertical asymptote of $C$ is $x=-2$,

Question

The curve $C$ has equation $y=\frac{{{x}^{2}}+ax+b}{c-x}$. The vertical asymptote of $C$ is $x=-2$, and the coordinates of the turning points are $\left( -4,2 \right)$ and $\left( 0,-6 \right)$. Find the values of $a$, $b$ and $c$. [3] Sketch $C$, stating the equations of the asymptotes. [2] By drawing an appropriate graph on the sketch of $C$, find the range of values of $k$$\left( k>0 \right)$ such that the equation ${{\left( x+4 \right)}^{2}}+{{\left( \frac{{{x}^{2}}+ax+b}{c-x} \right)}^{2}}={{k}^{2}}$ has no real roots. [2] When $C$ undergoes a translation in the direction of $y$-axis by $p$ units, $C$ intersects the line $y=-1$ at one negative value of $x$. State the set of values for $p$. [2]

Accepted Answer

(i) $a=6$, $b=12$, $c=-2$ (ii) Asymptotes $x=-2$ and $y=-x-4$ (iii) $05$

2013 PJC P2 Q5

2013 PJC P2 Q5

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