Intersections with Lines

Where a line meets a circle, substitute the line into the circle equation and read the resulting quadratic. The engine is identical to line–quadratic intersection; only the geometry vocabulary changes.

The routine

Where does $y = x - 1$ meet $x^2 + y^2 = 25$? Substitute (brackets intact): $x^2 + (x - 1)^2 = 25$ Expand, collect: $2x^2 - 2x - 24 = 0 \to x^2 - x - 12 = 0 \to (x - 4)(x + 3) = 0$ $x = 4 \to y = 3$; $x = -3 \to y = -4$ ($y$ from the line, as always) The line cuts the circle at $(4, 3)$ and $(-3, -4)$, a chord.

The substitution line with brackets is the M mark; pairing the coordinates correctly carries the finish (the pairing discipline).

The three verdicts, by discriminant

After collecting to a quadratic, $b^2 - 4ac$ delivers the geometry without solving:

• > 0, two points: the line is a chord (or secant)
• = 0, one repeated point: the line is a tangent
• < 0, no real points: the line misses the circle

“Show that $y = 2x + 10$ is a tangent to the circle…” $\to$ substitute, collect, show $b^2 - 4ac = 0$, state the conclusion. “Find $k$ so the line misses the circle” $\to$ discriminant $< 0 \to$ a quadratic inequality in $k$. The translation line (“tangent $\Rightarrow b^2 - 4ac = 0$”) is creditable working, write it before computing.

The geometric alternative (and when it wins)

Tangency can also be shown via perpendicular distance from the centre = radius, geometrically elegant, and occasionally faster when the circle is in completed form already. But the discriminant route reuses machinery you’ve drilled since quadratics, has no new formula to misremember, and earns identical credit. Default to it; mention the other only if it’s clearly shorter.

Chord follow-ups recycle coordinate tools: chord length = distance between the two points (exact surd); midpoint of the chord lies on the perpendicular from the centre.

Common mistakes

• Brackets dropped substituting $(x - 1)^2$, the topic’s costliest slip
• $a, b, c$ read before collecting to "$= 0$"
• $y$-values from the circle instead of the line (invites false pairings)
• Tangency claimed from one solution found, without the discriminant/repeated-root shown
• Full solving when only the verdict (or $k$) was asked

Full topic context: Circle Geometry notes.