Question 3
Curve C has equation $y=\frac{ax+3}{1-2x}$ where $x\neq\frac{1}{2}$ and $a$ is a constant.
The asymptote to C that is parallel to the $x$-axis has equation $y=4$.
(a) Find the value of $a$.
(b) Write down the equation of the asymptote to C that is parallel to the $y$-axis.
(c) Find the coordinates of the point where C crosses
(i) the $x$-axis,
(ii) the $y$-axis.
(d) Using the axes below, sketch $C$, showing clearly the asymptotes and the coordinates of the points where $C$ crosses the coordinate axes.
Step (a) Find the value of $a$
We are given the curve equation $y=\frac{ax+3}{1-2x}$, where $x\neq\frac{1}{2}$. The curve has an asymptote that is parallel to the $x$-axis with the equation $y=4$.
As $x\to\infty$, the value of $y$ approaches the horizontal asymptote. To find the asymptote, we consider the behavior of $y$ for large values of $x$. For large $x$, the numerator $ax+3$ behaves like $ax$ and the denominator $1-2x$ behaves like $-2x$. Therefore, we have:
$ y\approx\frac{ax}{-2x}=-\frac{a}{2}. $
For the horizontal asymptote to be $y=4$, we set:
$ -\frac{a}{2}=4. $
Solving for $a$, we get:
$ a=-8. $
Thus, the value of $a$ is $\boxed{-8}$.
Step (b) Write down the equation of the asymptote to $C$ that is parallel to the $y$-axis
Next, we need to find the asymptote that is parallel to the $y$-axis. Vertical asymptotes occur when the denominator of the function becomes zero. The denominator of the given curve is $1-2x$, so we set it equal to zero to find the $x$-value of the vertical asymptote:
$ 1-2x=0 \quad\Rightarrow\quad x=\frac{1}{2}. $
Therefore, the equation of the vertical asymptote is:
$ x=\frac{1}{2}. $
Step (c) Find the coordinates of the point where $C$ crosses
(i) The $x$-axis
The curve crosses the $x$-axis where $y=0$. Setting $y=0$ in the equation of the curve:
$ 0=\frac{ax+3}{1-2x}. $
This implies that the numerator must be zero:
$ ax+3=0 \quad\Rightarrow\quad x=-\frac{3}{a}. $
Substituting $a=-8$, we get:
$ x=-\frac{3}{-8}=\frac{3}{8}. $
Thus, the point where the curve crosses the $x$-axis is $\left(\frac{3}{8},0\right)$.
(ii) The $y$-axis
The curve crosses the $y$-axis where $x=0$. Substituting $x=0$ into the equation of the curve:
$ y=\frac{a(0)+3}{1-2(0)} =\frac{3}{1} =3. $
Thus, the point where the curve crosses the $y$-axis is $(0,3)$.
Step (d) Sketch of the Curve
- The curve has a horizontal asymptote at $y=4$.
- The curve has a vertical asymptote at $x=\frac{1}{2}$.
- The curve crosses the $x$-axis at $\left(\frac{3}{8},0\right)$ and the $y$-axis at $(0,3)$.
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