Mathematics 9709/13 — October/November 2016

Walkthrough

We start by treating the curve and the line as two equations in $x$ and $y$. To find their intersection points, we set the two expressions for $y$ equal. This is the standard way to solve simultaneous equations where one is a quadratic.

After equating, we get a quadratic equation in $x$: $kx^2 - 4x + k = 0$. The number of intersection points is the number of real solutions of this quadratic. If the discriminant is positive, there are two intersections; if it is zero, there is one; if it is negative, there are none.

Since the question asks for values of $k$ where they do not meet, we need the discriminant to be negative. We compute $b^2 - 4ac$ with $a=k$, $b=-4$, $c=k$.

The discriminant is $16 - 4k^2$. Setting this less than zero gives $k^2 > 4$. Solving this inequality gives two separate intervals: $k > 2$ or $k < -2$.

Finally, we note that $k=0$ would make the equation linear, but it would still have a solution, so it is not part of the answer. The inequality already excludes it.

Key Takeaways

• Intersections between a curve and a line are found by equating their expressions.
• The discriminant of the resulting quadratic determines how many intersections exist.
• "Do not meet" means the discriminant is negative, not positive or zero.
• Solving $k^2 > 4$ produces two separate intervals, not a single interval.

Common Mistakes

• Using $\Delta > 0$ instead of $\Delta < 0$; this would give values where the curve and line meet twice.
• Forgetting to rearrange the equation into the form $ax^2 + bx + c = 0$ before reading off $a$, $b$, $c$.
• Writing $k^2 > 4$ as $k > 2$ only, missing $k < -2$.
• Not showing the elimination step; the mark scheme requires the method to be shown for the first mark.

Things to Be Careful About

• The coefficient of $x^2$ is $k$, not $1$, so the discriminant is $(-4)^2 - 4(k)(k)$, not $(-4)^2 - 4k$.
• If $k = 0$, the equation becomes linear, however, it still has a solution, so it cannot be in the answer.
• The final answer must be written as two separate inequalities or as a union of intervals, e.g. $(-\infty, -2) \cup (2, \infty)$.