6.3PureStretch

Intersections of Straight Lines and Circles

A line can cut a circle twice, touch it once (a tangent), or miss it. Substituting the line into the circle gives a quadratic, and its discriminant tells you which of the three cases you are in.

30 min Video by Zeeshan Zamurred Circles

Edexcel AS Level Maths: 6.3 Intersections of Straight Lines and CirclesWatch the full walkthrough before the notes below.

Open on YouTube

What you'll be able to do

Find intersection points of a line and a circle
Use substitution to form a quadratic
Use the discriminant to classify line–circle position
Recognise the tangent condition

Substitute the line into the circle

Rearrange the line to $y = \dots$ and substitute into the circle equation. This produces a quadratic in $x$ ; its solutions are the $x$ -coordinates of the intersection points.

1Substitute:

x^{2} + x^{2} = 8 \Rightarrow 2 x^{2} = 8 \Rightarrow x^{2} = 4

2So

x = \pm 2

, and

y = x

Answer

(2, 2)

and

(- 2, - 2)

Two, one or none

The discriminant of that quadratic classifies the line: two points (the line is a secant), one repeated point (a $tangent$ ), or none (the line misses the circle).

b^{2} - 4 a c > 0 (2), = 0 (tangent), < 0 (0)

Same discriminant rule, now applied to a line and a circle.

Tip — “The line is a tangent to the circle” ⇔ the resulting quadratic has discriminant 0.

Tangent problems

A common question: find the value of a constant that makes a line a tangent. Substitute, form the quadratic, set its discriminant to zero, and solve.

Formula recap

y = m x + c into (x - a)^{2} + (y - b)^{2} = r^{2}

Substitute line into circle.

b^{2} - 4 a c = 0 \Leftrightarrow tangent

The tangent condition.

Common mistakes to avoid

Solving the line and circle separately instead of substituting.

Substitute the line into the circle to get a single quadratic.

Thinking a line must always cross a circle twice.

It can cross twice, touch once (tangent), or miss entirely.

Key takeaways

Substitute the line into the circle to form a quadratic.
Discriminant > 0: two points; = 0: tangent; < 0: no intersection.
For tangent problems, set the discriminant to zero and solve.

Test yourself

Ready to lock in Intersections of Straight Lines and Circles? Pick a mode and earn XP & Dobloons.