Where a Line Meets a Circle

Line-meets-circle is the same idea as line-meets-curve: substitute the line into the circle to get one quadratic, solve it, then read off both coordinates of each point. The only care needed is pairing each $x$ with the correct $y$ using the line.

Find the coordinates of the points where the line $y = x + 1$ meets the circle $x^2 + y^2 = 25$. [5]

The working

Step 1, substitute $y = x + 1$ into the circle: $x^2 + (x + 1)^2 = 25 \quad \text{(M1)}$

Step 2, expand and simplify to a quadratic in $x$: $x^2 + x^2 + 2x + 1 = 25 \;\Rightarrow\; 2x^2 + 2x - 24 = 0 \;\Rightarrow\; x^2 + x - 12 = 0 \quad \text{(A1)}$

Step 3, factorise and solve: $(x + 4)(x - 3) = 0 \;\Rightarrow\; x = -4 \quad \text{or} \quad x = 3 \quad \text{(M1)}$

Step 4, find each $y$ from the line $y = x + 1$: $x = -4 \Rightarrow y = -3, \qquad x = 3 \Rightarrow y = 4$

So the line meets the circle at $(-4, -3)$ and $(3, 4)$. (A1, A1)

Where the marks are won and lost

• Substitute into the circle, solve for $x$, then use the line for $y$. Finding $y$ from the circle would force you to choose a sign and risks pairing the wrong coordinates.
• Expanding $(x + 1)^2$ needs the middle term $2x$. Dropping it gives the wrong quadratic.
• Dividing through by the common factor $2$ (to get $x^2 + x - 12 = 0$) makes the factorising clean, a small step that saves errors.

Common mistakes

• Solving for $y$ from $y^2 = 25 - x^2$ and mismatching the pairs.
• Expanding $(x+1)^2$ as $x^2 + 1$.
• Giving only the $x$-values and forgetting the $y$-coordinates.

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