Points of Intersection

“Find the points where the line meets the curve” is simultaneous equations wearing geometry: each intersection point is one $(x, y)$ pair satisfying both equations. The algebra is identical; what changes is what the exam does with the points afterwards.

Solve, pair, and say where

Find the points of intersection of $y = 2x - 1$ and $y = x^2 - 4x + 8$. $x^2 - 4x + 8 = 2x - 1 \to x^2 - 6x + 9 = 0 \to (x - 3)^2 = 0 \to x = 3$ (repeated) $y = 2(3) - 1 = 5 \to$ the line touches the curve at $(3, 5)$, it is a tangent.

A repeated root isn’t a nuisance, it’s information: one repeated solution means tangency, and saying “the line is a tangent” is often a mark. Two roots $\to$ two crossing points; no real roots $\to$ no intersection (the discriminant view).

What the points get used for

0606 rarely stops at the points. The standard follow-ups, each leaning on a different topic:

• Length of the chord $AB$ between the two points → distance formula, exact surd answer on Paper 1
• Midpoint of $AB$ → midpoint formula; sometimes feeding a perpendicular bisector
• Area between the line and curve → the intersection $x$-values become the integration limits
• Circle questions → where a line cuts a circle, with the same substitute-collect-solve engine (circle intersections)

Because the points feed later parts, errors propagate, which is why the back-substitute-into-the-linear habit and paired presentation matter beyond their own marks: a mispaired point poisons the chord length two parts later.

Labelling for multi-part questions

When parts (ii) and (iii) will reuse the points, label them in part (i): ”$A(1, 1)$, $B(5, 9)$”. It costs nothing, prevents self-confusion, and lets the examiner follow transferred values, protecting follow-through marks if an early slip occurred.

Common mistakes

• $y$-values computed from the curve instead of the line
• Repeated roots not interpreted (tangency unstated)
• Points unpaired or unlabelled, then misused downstream
• Exact surd coordinates rounded before a later part needed them exact
• The follow-up ignored when “hence” pointed straight at the points

Full topic context: Simultaneous Equations notes.