Two graphs

Tangent, meet, don't meet

In 5 of 20 recent exams
2% of the marks
Paper 1, no calculator
4 to 6 marks

How it is set in the exam

Paper 1 (no calculator), 4–6 marks

Spot it in the exam

Exam phrases to spot (4)

“the line … is a tangent to the curve …”
“meets the curve at two points”
“does not meet the curve”
“the curves intersect at exactly one point”

In disguise (6)

"the line and the curve have exactly one point in common" (= tangent)
"show that the line never intersects the curve"
two parabolas: set them equal and the $x^2$ terms may cancel, leaving a linear equation (exactly one solution, whatever the unknown is, unless the $x$ terms cancel too)
a sloping line and a rational function $\frac{ax + b}{cx + d}$ : multiply by the denominator first, and the equation is a quadratic. A root at the excluded $x$ is not a meeting point; a horizontal line gives a linear equation instead (Where rational functions hide)
two log curves: write each side as a single log with the log laws (e.g. $\ln x + \ln(x + 2) = \ln\left(x(x + 2)\right)$ ), drop the $\ln$ , and the meeting equation is a quadratic with the unknown in it. Then $\Delta$ as usual; a root counts only if every log argument is positive there.
a condition on where they meet (say, two meeting points with $x \ge 0$ ) when the line passes through a point already on the curve: that point's $x$ is one root, so $\Delta$ is a perfect square and decides nothing. Factor out the known root and put the condition on the other one (compare step 5 of Find k from the number of roots).

Looks like it, but isn't (4)

tangent at a given point (for example "at $x = 3$ "): substitute the point, or use calculus (Tangent or normal at a given point)
a horizontal line $y = c$ meeting a parabola once: the line goes through the vertex, so use Find the vertex
a tangent to a cubic, trig or exponential curve: substituting the line gives no quadratic, so there is no $\Delta$ , and a tangent can meet such a curve again. Use the derivative (Find the point of contact)
how many times two graphs meet when setting them equal gives no quadratic (an exponential and a parabola, say): sketch both and count (Solve graphically

Worked example

Tangent with an unknown gradient

Paper 1 (no calculator)

Question

The line $y = mx - 1$ is a tangent to the curve $y = x^2 + 3$ .
(a) Find the possible values of $m$ . [5]
(b) For the positive value of $m$ , find the point of contact. [2]

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See it move: Tilt the line: twice, once or never

$m = 2.5$ : $x^{2} + 3 = 2.5x - 1$ gives $x^{2} - 2.5x + 4 = 0$ , with $\Delta = m^{2} - 16 = -9.75$ : no real roots: the line misses the curve.
"Tangent" with $m$ unknown and no point given means the discriminant route: set the two equal and bring everything to one side, $x^{2} - mx + 4 = 0$ . The line pivots about $(0, -1)$ .
One meeting point is a repeated root, so $\Delta = m^{2} - 16 = 0$ and $m = \pm 4$ . Between them $\Delta < 0$ and the line misses; beyond them it crosses twice.
(b) For $m = 4$ the repeated root is $x = 2$ , and $y = 4(2) - 1 = 7$ : the point of contact is $(2, 7)$ . A point needs both coordinates.
The other route: the curve's gradient is $2x$ , so $m = 2x$ at the contact point, which is on both graphs: $x^{2} + 3 = 2x^{2} - 1$ , $x = \pm 2$ and $m = 4$ there. Both routes are valid.

Steps

Set the two expressions equal.
Rearrange everything to one side: $ax^2 + bx + c = 0$ .
Apply the discriminant condition and solve for the unknown (Find k from the number of roots).
If asked for the point of contact, substitute the value back and solve (it will be a repeated root). Then find the y-coordinate too.
If a later part fixes the unknown and asks where the graphs meet, put the value into the quadratic from step 2 and solve it exactly (formula or completing the square). On the answers are surds, and so is a distance between them.

Common mistakes

Not rearranging fully.
The discriminant only works on " $\ldots = 0$ ".
Losing the brackets on $b$
$b = 1 - k$ , so $b^2 = (1-k)^2$ .
Using $\Delta = 0$ for "meets".
"Meets" alone means at least once ( $\Delta \geq 0$ ); "meets at two points" means $\Delta > 0$ .

More mistakes (3)

Assuming "tangent" means calculus.
For a parabola it is valid, but slow when no point is given: use the discriminant. (For a cubic, trig or exponential curve it's the other way round: only the derivative works.)
Forgetting the y-coordinate
of the point of contact. Substitute $x$ back into either equation.
Decimals where "exact" is asked
$x = 2 \pm \sqrt{3}$ stays in surds on . $0.268$ and $3.73$ are not exact, and a distance of $3.46$ loses the final mark.

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