The curve $C$ has equation $y=\frac{{{x}^{2}}+5}{\lambda x+1}$, where $\lambda $ is a non-zero const

Question

The curve $C$ has equation $y=\frac{{{x}^{2}}+5}{\lambda x+1}$, where $\lambda $ is a non-zero constant. Obtain the equations of the asymptotes of $C$. Show that $C$ has exactly two points at which $\frac{\mathrm{d}y}{\mathrm{d}x}=0$. Draw a sketch of $C$ to illustrate the case $\lambda >0$, and a separate of $C$ to illustrate the case $\lambda Find the set of values of $\lambda $ for which the line $y=4x$ and $C$ has at least one point in common.

Accepted Answer

(i) OA: $y = \frac{x}{\lambda} - \frac{1}{\lambda^2}$, VA: $x = -\frac{1}{\lambda}$. (iv) $\lambda \ge \frac{1}{20}$

N1999 P1 Q1 (FM)

N1999 P1 Q1 (FM)

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