An intersection found by algebra must still obey every condition in the question. Write the restriction down first, then test each answer against it as a last step.
This lesson is part of geometry problems with several possible approaches. It follows choosing a coordinate method rather than a vector method.
What is the order of steps?
- Underline the restriction in the question.
- Solve the equations to find every algebraic intersection.
- Test each answer against the restriction, and reject those that fail.
- State which answer remains, or that none does.
Worked example 1: a segment and two lines
The segment AB joins A(1, 1) and B(3, 5). The line through AB is y = 2x − 1, and on the segment x runs from 1 to 3.
Line M: x + y = 8. Substitute: x + 2x − 1 = 8, so x = 3 and y = 5. The point is (3, 5), which is B. It lies on the segment, at its end.
Line N: x + y = 14. Substitute: 3x − 1 = 14, so x = 5 and y = 9. The point (5, 9) is on the line AB, but x = 5 is outside 1 ≤ x ≤ 3. It is not on the segment.
Worked example 2: a line and a curve
The line y = 2x − 3 meets the curve y = x² − 4x + 5. Setting them equal: x² − 6x + 8 = 0, so (x − 2)(x − 4) = 0 and x = 2 or x = 4.
The points are (2, 1) and (4, 5). Check (2, 1): the curve gives 4 − 8 + 5 = 1 and the line gives 4 − 3 = 1. Check (4, 5): the curve gives 16 − 16 + 5 = 5 and the line gives 8 − 3 = 5.
If the question says “P is the point nearer to the y-axis”, P = (2, 1), because its x value is smaller.
The mistake that costs marks
A common slip is to give every algebraic answer and skip the restriction. Take line N above.
| Answer given | Restriction tested? | Result |
|---|---|---|
| Line N meets segment AB at (5, 9) | No | Wrong: the point is beyond B |
| Line N meets line AB at (5, 9), but not the segment | Yes | Correct: no intersection with the segment |
The algebra is identical in both answers. Only the final check separates a full mark from none.
Check yourself
The segment PQ joins P(0, 0) and Q(4, 8), with line y = 2x and 0 ≤ x ≤ 4. Does the line x + y = 15 meet the segment?
Answer
Substitute y = 2x: x + 2x = 15, so x = 5 and y = 10. The lines meet at (5, 10).
The segment has x between 0 and 4, and 5 is outside that range. So the line does not meet the segment. The point (5, 10) lies on the line PQ extended beyond Q.
What to study next
Test yourself with the geometry problems practice set, and read recovering a parameter when a line touches a curve.
The word-problem structure worksheet helps you underline restrictions. A teacher can make the check a habit in online one-to-one Additional Mathematics tuition.