Does a Line Intersect a Circle?

Whether a line cuts, touches, or misses a circle is decided by the discriminant of the combined equation, no sketch needed. Substitute the line into the circle, then read the sign of $b^2 - 4ac$.

Determine whether the line $y = x + 6$ intersects the circle $x^2 + y^2 = 16$. [5]

The working

Step 1, substitute $y = x + 6$ into the circle: $x^2 + (x + 6)^2 = 16 \quad \text{(M1)}$

Step 2, expand and collect to a quadratic in $x$: $x^2 + x^2 + 12x + 36 = 16 \;\Rightarrow\; 2x^2 + 12x + 20 = 0 \;\Rightarrow\; x^2 + 6x + 10 = 0 \quad \text{(A1)}$

Step 3, compute the discriminant with $a = 1$, $b = 6$, $c = 10$: $b^2 - 4ac = 6^2 - 4(1)(10) = 36 - 40 = -4 \quad \text{(M1, A1)}$

Step 4, interpret. Since $b^2 - 4ac = -4 < 0$, the quadratic has no real roots, so the line does not intersect the circle, it misses it entirely. (A1)

Where the marks are won and lost

• Expand $(x + 6)^2$ fully to $x^2 + 12x + 36$ (the middle term matters), then collect. Dividing by $2$ simplifies to $x^2 + 6x + 10 = 0$.
• The sign of the discriminant is the answer: negative means miss, zero means tangent, positive means two intersections. State the interpretation, not just the number.
• No need to solve the quadratic, the discriminant alone answers “does it intersect”.

Common mistakes

• Expanding $(x + 6)^2$ as $x^2 + 36$ (dropping $12x$).
• Computing the discriminant but not interpreting its sign.
• Concluding “tangent” for a negative discriminant (that’s the zero case).

Full method: Intersections with Lines notes. Topic home: Circle Geometry pillar.