Mathematics 9709/11 — October/November 2020

Walkthrough

We are comparing a straight line and a quadratic curve. To find whether they intersect, we set their $y$-values equal. This produces a quadratic equation in $x$.

If that quadratic has real solutions, there are one or two points where the line and curve meet. If it has no real solutions, the line and curve never meet. A quadratic has no real solutions exactly when its discriminant is negative.

Here the equation is

which rearranges to

We identify $a = 2$, $b = -m$, $c = 8$. Substituting into $b^2 - 4ac$ gives $m^2 - 64$. Demanding this be negative gives $m^2 < 64$, so $-8 < m < 8$.

Key Takeaways

The intersection of a line and a curve is found by eliminating $y$. The discriminant determines how many intersections there are: positive means two intersections, zero means one tangent point, negative means none. Solving $m^2 < k^2$ gives $-k < m < k$.

Common Mistakes

• Failing to rearrange to all terms on one side before reading $a$, $b$, $c$.
• Using $b = m$ instead of $b = -m$; however, because the term is squared, it does not affect the final discriminant.
• Forgetting the factor $2$ and $8$ in $4ac$, leading to an incorrect discriminant.
• Using the wrong inequality sign: no intersection requires discriminant $< 0$, not $> 0$.
• Including the endpoints incorrectly. At $m = \pm 8$ the discriminant is zero, so the line is tangent to the curve and touches it, so strictly the endpoints are excluded.

Things to Be Careful About

The question asks for the values for which the line and curve do not meet. A zero discriminant means a tangent, which is a meeting at exactly one point, so the inequality must be strict: $-8 < m < 8$. The answer may also be written as the interval $(-8, 8)$.