The curve $C$ has equation $y=\frac{{{x}^{2}}+ax+4}{x+b}$. It is given that $C$ has vertical asympto

Question

The curve $C$ has equation $y=\frac{{{x}^{2}}+ax+4}{x+b}$. It is given that $C$ has vertical asymptote $x=-1$ and a stationary point at $x=2$. Determine the values of $a$ and $b$. [3] Find the equation of the other asymptote of $C$. [1] Prove, using an algebraic method, that $C$ cannot lie between two values (to be determined). [4] Draw a sketch of $C$, showing clearly any axial intercepts, asymptotes and stationary points. [3] Deduce the number of real roots of the equation $\left( 4-{{x}^{2}} \right){{\left( x+1 \right)}^{2}}={{\left( {{x}^{2}}-4x+4 \right)}^{2}}$. [2]

Accepted Answer

$a = -4, \; b = 1; \quad y = x - 5; \quad -12 < y < 0; \quad \text{2 solutions}$

2007 HCI P1 Q13

2007 HCI P1 Q13

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