The curve $C$ has equation $y=\frac{{{x}^{2}}+px+3}{x+q}$, where $p$ and $q$ are non-zero constants

Question

The curve $C$ has equation $y=\frac{{{x}^{2}}+px+3}{x+q}$, where $p$ and $q$ are non-zero constants and $x
e -q$. It is given that the asymptotes of $C$ are $y=x+3$ and $x=2$. Find the values of $p$ and $q$. Hence sketch $C$, stating the coordinates of any axial intercepts and turning points. [4] Use the values of $p$ and $q$ that you have found in part (i) to answer part (ii). By drawing a suitable curve on the same diagram in part (i), find the range of values of $b$, where $b$ is a positive constant, such that there are $2$ real roots to the equation ${{\left( x-5 ight)}^{2}}+b{{\left( \frac{{{x}^{2}}+px+3}{x+q} ight)}^{2}}=1$. [2]

Accepted Answer

(i) $p=1,q=-2$; (ii) $0<b<\frac1{121}$

2021 DHS P1 Q3

2021 DHS P1 Q3

Video Solution:

Solution:

Solution locked