To find where a line meets a circle, replace one variable in the circle equation with the expression from the line, solve the quadratic that results, then find each partner coordinate from the line. This appears in Additional Mathematics whenever a question asks for the points of intersection of a line and a circle.
The skill sits inside simultaneous linear and nonlinear models, and the circle equations themselves are explained in circle coordinate methods.
Why does substitution work?
A point on both the line and the circle satisfies both equations at once. So if the line says y = x + 1, then at any shared point y and x + 1 are the same number. Writing x + 1 wherever y appears in the circle equation keeps the statement true and leaves one unknown.
What remains is a quadratic in x. A quadratic can have two roots, one repeated root or none, which matches a line crossing, touching or missing a circle.
How do you do it, step by step?
- Rearrange the line so one variable is on its own, usually y = mx + c.
- Substitute that expression into the circle, putting it inside brackets.
- Expand the bracket fully and collect everything on one side.
- Divide by any common factor to make the quadratic simpler.
- Solve the quadratic by factorising or the formula.
- Find each partner value using the line, not the circle.
- Check both points in both equations.
Step 6 uses the line because it gives exactly one partner for each x. The circle would offer two y-values for every x and invite errors.
Worked example
Find the points where the line y = 2x − 1 meets the circle x² + y² = 10.
Substitute: x² + (2x − 1)² = 10.
Expand: (2x − 1)² = 4x² − 4x + 1, so x² + 4x² − 4x + 1 = 10.
Collect: 5x² − 4x − 9 = 0.
Factorise: (5x − 9)(x + 1) = 0, so x = 9/5 or x = −1.
Partners from the line: when x = −1, y = 2(−1) − 1 = −3. When x = 9/5, y = 18/5 − 1 = 13/5.
Answer: (−1, −3) and (9/5, 13/5).
Check: (−1)² + (−3)² = 1 + 9 = 10. And (9/5)² + (13/5)² = 81/25 + 169/25 = 250/25 = 10. Both lie on the line as well, since 13/5 = 2(9/5) − 1.
The mistake to watch for
The most common slip is to treat the bracket as if the square goes through each term.
Mistaken working: x² + (2x − 1)² = 10 becomes x² + 4x² − 1 = 10.
The student squared 2x and squared 1 but dropped the middle term −4x.
That gives 5x² = 11, a neat-looking equation with the wrong answer and no sign of the second intersection. The fix is to write the bracket twice, (2x − 1)(2x − 1), and multiply all four products.
The line has a constant term, so a middle term must appear. A missing one is a warning sign.
Check yourself
Work these on paper, then open each answer.
1. Find where the line x + y = 7 meets the circle x² + y² = 25.
Show answer
y = 7 − x. Then x² + (7 − x)² = 25, so 2x² − 14x + 49 = 25, which gives 2x² − 14x + 24 = 0 and x² − 7x + 12 = 0. So (x − 3)(x − 4) = 0 and x = 3 or 4.
Partners: y = 4 and y = 3.
(3, 4) and (4, 3).
2. Find where y = x + 1 meets the circle (x − 2)² + (y − 1)² = 10.
Show answer
Substitute: (x − 2)² + x² = 10, since y − 1 = x. Expand: x² − 4x + 4 + x² = 10, so 2x² − 4x − 6 = 0 and x² − 2x − 3 = 0. Then (x − 3)(x + 1) = 0, so x = 3 or −1.
Partners: y = 4 and y = 0.
(3, 4) and (−1, 0). Check (3, 4): 1 + 9 = 10. Check (−1, 0): 9 + 1 = 10.
3. A student substitutes y = x + 3 into x² + y² = 29 and writes x² + x² + 9 = 29. What went wrong, and what should the line be?
Show answer
The square of (x + 3) was written as x² + 9, so the middle term 6x is missing. The correct line is x² + x² + 6x + 9 = 29, which gives 2x² + 6x − 20 = 0, then x² + 3x − 10 = 0, so x = 2 or x = −5.
Where this leads next
Next, see what happens when both equations are curved in eliminating a variable from a nonlinear pair, then test the module with the mixed practice set. The line and circle intersection explorer draws the picture so you can confirm your coordinates, and the non-calculator working trainer builds speed with exact fractions like 9/5.
Some students understand every line of a worked solution but still lose marks producing one unaided. That is where our teachers work in online one-to-one Additional Mathematics tuition.