A curve $C$ has equation $y=\frac{a{{x}^{2}}+bx+c}{x-d}$, where $a$, $b$, $c$ and \(d

Question

A curve $C$ has equation $y=\frac{a{{x}^{2}}+bx+c}{x-d}$, where $a$, $b$, $c$ and $d$ are constants. It is given that two of its asymptotes are $y=x+2$ and $x=1$. State the value of $d$, and show that $a=b=1$.[2] Using differentiation, find the range of values of $c$ such that the graph of $C$ contains two stationary points.[4] Use $c=14$ for the rest of the question. Sketch $C$, showing clearly the equations of asymptotes and the coordinates of the turning points.[3] State the maximum number of roots to the equation ${{k}^{2}}{{\left( x-5 \right)}^{2}}+{{\left( \frac{a{{x}^{2}}+bx+c}{x-d}-3 \right)}^{2}}={{k}^{2}}$, where $k>0$. Deduce the range of values of $k$ for the maximum number of roots to occur.[2]

Accepted Answer

(a) $d=1,\ a=b=1$ (shown) (b) $c>-2$ (c) Asymptotes $x=1$ and $y=x+2$; turning maximum $(-3,-5)$ and minimum $(5,11)$ (d) Maximum $2$ roots, for $k>8$

2023 CJC P1 Q6

2023 CJC P1 Q6

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