Line–Curve Intersection

Where a line meets a curve, their equations agree, so substitute one into the other and read the resulting quadratic. One routine answers every variant 0606 asks.

The routine

Find where $y = x + 1$ meets $y = x^2 - 3x + 6$. Substitute: $x^2 - 3x + 6 = x + 1$ (brackets intact if the line had them) Collect to zero: $x^2 - 4x + 5 = 0$… discriminant $16 - 20 = -4 < 0 \to$ they never meet.

Same setup, three possible endings:

1. Two solutions $\to$ the line crosses at two points: solve, then get $y$-values from the line’s equation, present as coordinate pairs (the pairing discipline).
2. One repeated solution $\to$ tangent: the line touches the curve.
3. No real solutions $\to$ no contact, and “show that the line does not meet the curve” wants the discriminant line and the verbal conclusion.

The tangent/find-k family

The exam’s favourite inversion: the geometry is given, the constant is unknown.

$y = 2x + k$ is a tangent to $y = x^2 + 6x + 11$. Find $k$. $x^2 + 6x + 11 = 2x + k \to x^2 + 4x + (11 - k) = 0$ Tangent $\Rightarrow$ equal roots $\Rightarrow$ $b^2 - 4ac = 0$: $16 - 4(11 - k) = 0 \to 4k = 28 \to$ $k = 7$

“Find the range of values of $m$ for which $y = mx + 3$ cuts the curve in two points” runs identically but ends in a quadratic inequality in $m$. The geometric word (“tangent”, “cuts twice”, “does not meet”) translates directly to a discriminant condition, write the translation as its own line; it’s the method mark.

Read what’s wanted before solving

If coordinates are wanted, solve fully. If only whether/when contact happens, go straight to the discriminant, solving for points you don’t need burns Paper-time the marks budget can’t spare.

Common mistakes

• Substituting from the quadratic into the line (backwards, always sub the line in)
• $a$, $b$, $c$ identified before collecting to "$= 0$"
• $y$-values found from the curve (more algebra, more slips) instead of the line
• Tangent questions fully solved when only $k$ was asked
• The conclusion sentence omitted on “show that” variants

Full topic context: Quadratic Functions notes · the same idea with circles: intersections with lines.