1.12: Past-paper question 9

A curve $\displaystyle C$ has equation $\displaystyle y = \frac{3 - 2x}{x + 6}$ where $\displaystyle x \neq -6$

(a) Write down an equation of the asymptote to $\displaystyle C$ that is parallel to the

(i) $\displaystyle x$-axis (ii) $\displaystyle y$-axis

(b) Find the coordinates of the point where $\displaystyle C$ crosses the

(i) $\displaystyle x$-axis (ii) $\displaystyle y$-axis

(c) Using the axes opposite, sketch the graph of $\displaystyle C$, showing clearly its asymptotes and the coordinates of the points where $\displaystyle C$ crosses the coordinate axes.

(d) Show that the gradient of the tangent to $\displaystyle C$ is always negative.

A tangent to $\displaystyle C$ has equation $\displaystyle y = -\frac{3}{5}x + k$ where $\displaystyle k > 0$

(e) Find the value of $\displaystyle k$