How much this section is worth
The "Geometry and Algebra" module in the official syllabus covers plane analytic geometry: the equations and properties of lines, circles, ellipses, hyperbolas and parabolas.
We counted 7 past papers with 333 questions in total, with each question assigned to a single knowledge point according to its main test point. The Geometry and Algebra module accounted for 94 questions, or 28%; of these, the 8 knowledge points on lines and conic sections came to 75 questions, about 23% of all 333.
A separate set of 23 questions covers the "plane rectangular coordinate system"; in our classification it belongs to the functions chapter and is the foundation for studying this section.
Number of past-paper questions for the 8 knowledge points
- Lines (35 questions): 9 on angle of inclination and slope, 8 on equations of lines, and 12 on parallel and perpendicular lines, all three of which appeared in all 7 papers; 6 on intersection points, which appeared in 6 papers.
- Conic sections (40 questions): 14 on circles and 10 on ellipses, both of which appeared in all 7 papers; 10 on parabolas and 6 on hyperbolas, which appeared in 6 papers.
Suggested review order
You can review lines first and conic sections second.
- Learn lines first: slope k = tanθ = (y₂ − y₁)/(x₂ − x₁); point-slope form y − y₀ = k(x − x₀); general form Ax + By + C = 0.
- Parallel and perpendicular: when two non-coincident lines both have a slope, parallel ⇔ k₁ = k₂ and perpendicular ⇔ k₁·k₂ = −1.
- Distance from a point to a line: d = |Ax₀ + By₀ + C| / √(A² + B²).
- Circles: (x − a)² + (y − b)² = r², with center (a, b) and radius r; to determine the position of a line relative to a circle, compare the distance d from the center to the line with the radius r.
- Ellipses: x²/a² + y²/b² = 1 (a > b > 0), c² = a² − b², eccentricity e = c/a.
- Parabolas: y² = 2px (p > 0), focus (p/2, 0), directrix x = −p/2.
- Hyperbolas: x²/a² − y²/b² = 1, c² = a² + b², asymptotes y = ±(b/a)x.
Easy-to-confuse points
- For an ellipse c² = a² − b², whereas for a hyperbola c² = a² + b²; do not mix them up.
- A vertical line (x = constant) has no slope, so it needs to be considered separately.
- First check which type of curve the equation represents and which axis the focus lies on, then substitute into the formulas.
Next step
Start with the 15-minute diagnostic to see how well you have mastered this section, then go to the Learning Center to practice by knowledge point.