Equation & inequalities: Intersection(s) between line and curve
Deduce number of intersections between line and curve using discriminant
General cases:
1. If line meets curve at two points, then $b^2 - 4ac$ $ > 0 $
2. If line meets curve once or is tangent to the curve, then $b^2 - 4ac$ $ = 0 $
3. If line meets curve once or twice, then $b^2 - 4ac$ $ \ge 0 $
4. If line does not meet the curve, then $b^2 - 4ac$ $ < 0 $
Questions
Find coordinates of the point(s) of intersection between line and curve
Solve simultaneous equations (using equation of line and equation of curve) by substitution method.
The solutions to the simultaneous equations represent the coordinates $(x, y)$ of the point of intersection(s). For example, if one of the solutions is $x = 2, y = 3$, then the coordinate of the point is $(2, 3)$.
1. Find the coordinates of the points of intersection of the curve $x^2 = xy + 12$ and the line $2y + x - 6 = 0$.
(from think! Workbook A Worksheet 2C)
Answer: $ (4, 1), (-2, 4) $
Form simultaneous equations to solve real-life problem
2. A cylinder of wooden block has base radius $r$ cm, height $h$ cm and a total surface area of $32\pi$ cm2.
(from Addition Maths 360 Ex 1.1)
(i) Show that $r^2 + hr = 16$
(Note: In exams, if you can't obtain the equation, you can still use it to do part ii)
(ii) Given that its height is $4$ cm more than its base radius, find the value of $r$ and of $h$.
Answer: (ii) $ r = 2, h = 6 $
Using discriminant
Steps:
- Substitute equation of line into equation of curve (or vice versa) to form a quadratic equation ($ax^2 + bx + c = 0$)
- Find the discriminant $b^2 - 4ac$
- Apply the relevant conditions for the discriminant
3. The equation of a curve is $y = kx^2 + 2kx + k$, where $k$ is a constant. Find the range of values of $k$ such that the line $y = 1 - x$ meets the curve.
Answer: $ k \ge -{1 \over 8} $
4. The equation of a curve is $y = 2x^2 - kx - 1$, where $k$ is a constant, and the equation of a line is $y + 3x = 10$. Show that, for all values of $k$, the line intersects the curve at two distinct points.
(from Think A Maths Workbook Worksheet 2C
Past year O level questions
| Year & paper | Comments |
|---|---|
| 2025 P1 Question 2 | Discriminant: Curve is completely below line |
| 2025 P1 Question 8 | (a) Discriminant: Line meets curve at two points (b) Discriminant: Line is tangent to curve |
| 2024 P1 Question 4 | Find coordinates of the point(s) of intersection between line and curve, then find the midpoint of the two points |
| 2024 P2 Question 2 | Discriminant: Line meets curve at two distinct points |
| 2023 P1 Question 1 | Discriminant: Line is tangent to the curve |
| 2023 P1 Question 8 | Find coordinates of the point(s) of intersection between line and curve |
| 2022 P1 Question 1b | Find coordinates of the point(s) of intersection between line and curve, then find the midpoint of the two points |
| 2022 P1 Question 9b | Discriminant: Line is tangent to the curve |
| 2021 P1 Question 2 | Solve simultaneous equations |
| 2021 P2 Question 5b, c | Discriminant: Line is tangent to the curve |
| 4049 Specimen P2 Question 4a | Discriminant: Line does not meet curve |
| 2020 P2 Question 2a | Solve simultaneous equations |
| 2020 P2 Question 2c | Discriminant: Line is tangent to the curve |
| 2019 P1 Question 2 | Discriminant: Line does not meet curve |
| 2018 P2 Question 9i | Find coordinates of the point(s) of intersection between line and curve |
| 2017 P2 Question 6 | Discriminant: Line is tangent to the curve |
| 2016 P1 Question 1i | Find coordinates of the point(s) of intersection between line and curve |
| 2016 P1 Question 1ii | Discriminant: Show that line always intersects the curve at two points |
| 2014 P1 Question 9ii, iii | Discriminant: Line is tangent to the curve |
| 2012 P2 Question 8i | Discriminant: Line does not meet curve |
| 2011 P1 Question 6 | Form simultaneous equations to solve real-life problem |
| 2011 P2 Question 1ii | Discriminant: Line is tangent to the curve |
| 2009 P1 Question 6ii | Discriminant: Line is tangent to the curve |
| 2009 P1 Question 7 | Find coordinates of the point(s) of intersection between line and curve, then find the midpoint of the two points |
| 2008 P1 Question 9 | Form simultaneous equations to solve real-life problem (Link - Subscription required)) |
| 2007 P2 Question 1 | Discriminant: Line meets the curve at two points |
| 2007 P2 Question 3 | Find coordinates of the point(s) of intersection between line and curve, then find the distance between the two points |
| 2006 P2 Question 5 | Find coordinates of the point(s) of intersection between line and curve, then find the distance between the two points |
| 2006 P2 Question 7a | Discriminant: Line is tangent to the curve |
| 2005 P1 Question 2 | Discriminant: Line is tangent to the curve |
| 2005 P2 Question 2 | Find coordinates of the point(s) of intersection between line and curve, then find the midpoint of the two points |
| 2004 P1 Question 4 | Discriminant: Line meets the curve |
| 2003 P2 Question 1 | Find coordinates of the point(s) of intersection between line and curve, then find the midpoint of the two points |
| 2003 P1 Question 1 | Discriminant: Line does not meet curve |
| 2002 P1 Question 2 | Discriminant: Line is tangent to the curve |
| 2002 P1 Question 9 | Find coordinates of the point(s) of intersection between line and curve, then find the perpendicular bisector of the two points |