FPM(2023) January Paper 01 que 3 and solution

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 $(0,3)$.