Analytic Geometry and Conics
What this domain tests
Lines (slope, equations, parallelism, perpendicularity, distance), circles, and the three conics - parabola, ellipse and hyperbola - as loci with their foci, directrices, axes, asymptotes and eccentricity, plus tangency conditions found by the discriminant.
Core rules
- Line: y − y₁ = m(x − x₁) or Ax + By + Γ = 0, with slope m = −A/B and normal vector (A, B). Parallel: m₁ = m₂. Perpendicular: m₁m₂ = −1. Vertical lines have no slope - handle them separately.
- Distance from (x₀, y₀) to Ax + By + Γ = 0 is |Ax₀ + By₀ + Γ|/√(A² + B²).
- Circle: (x − a)² + (y − b)² = ρ² has centre (a, b), radius ρ. For x² + y² + Dx + Ey + F = 0 the centre is (−D/2, −E/2) and ρ² = D²/4 + E²/4 − F.
- Parabola y² = 4px: focus (p, 0), directrix x = −p; every point is equidistant from focus and directrix.
- Ellipse x²/a² + y²/b² = 1 with a > b > 0: foci (±γ, 0) where γ² = a² − b², eccentricity ε = γ/a < 1.
- Hyperbola x²/a² − y²/b² = 1: γ² = a² + b², asymptotes y = ±(b/a)x, eccentricity ε = γ/a > 1.
- Tangency to a curve: substitute the line into the curve and require the resulting quadratic to have a double root (Δ = 0).
Worked example
Find k so that the line y = x + k is tangent to the parabola y² = 8x.
Substitute: (x + k)² = 8x, so x² + 2kx + k² − 8x = 0, i.e. x² + (2k − 8)x + k² = 0.
Tangency means exactly one intersection, so the discriminant vanishes:
Δ = (2k − 8)² − 4k² = 4k² − 32k + 64 − 4k² = 64 − 32k = 0, giving k = 2.
The tangent is y = x + 2; the double root is x = 2, so the point of contact is (2, 4), and indeed 4² = 16 = 8·2.
Traps that cost marks
- Using Δ > 0 as the tangency condition (that gives two intersections) or forgetting to check the contact point.
- Mixing the relations γ² = a² − b² (ellipse) with γ² = a² + b² (hyperbola).
- Sign errors on the directrix: for y² = 4px it is x = −p, not x = p.
- Solving an intersection with a circle and discarding the ± from the square root too early.
- Treating the eccentricity of an ellipse as greater than 1, or leaving it as γ instead of γ/a.
Probability and Statistics
What this domain tests
Probability of combined events, conditional probability and independence, tree diagrams and counting; then descriptive statistics - mean, median, mode, range, variance and standard deviation, grouped data, linear transformations of data and the interpretation of the correlation coefficient.
Probability rules
- P(A ∪ B) = P(A) + P(B) − P(A ∩ B); P(A′) = 1 − P(A); mutually exclusive events have P(A ∩ B) = 0.
- Conditional probability: P(A|B) = P(A ∩ B)/P(B) for P(B) > 0.
- Independence: P(A ∩ B) = P(A)P(B), equivalently P(A|B) = P(A).
- Tree diagrams: multiply along a branch, add across branches.
- Counting: ordered arrangements n!/(n − k)!; unordered selections C(n, k) = n!/(k!(n − k)!).
Statistics rules
- Mean μ = Σx/n, or Σfx/Σf for grouped data; median is the middle value (or the mean of the two middle values); mode is the most frequent.
- Variance σ² = Σ(x − μ)²/n = Σx²/n − μ²; standard deviation σ = √σ², always non-negative.
- Linear change y = ax + b: ȳ = a·x̄ + b and σ_y = |a|·σ_x. Adding a constant shifts the mean but leaves the spread unchanged.
- Correlation coefficient r satisfies −1 ≤ r ≤ 1; its sign gives the direction of the linear relationship and |r| its strength. Adding a constant or multiplying by a positive constant leaves r unchanged.
Worked example
Given P(A) = 0.5, P(B) = 0.4 and P(A ∪ B) = 0.7, find P(A ∩ B), P(A|B) and decide whether A, B are independent.
Rearranging the addition rule: P(A ∩ B) = 0.5 + 0.4 − 0.7 = 0.2.
Conditional: P(A|B) = 0.2/0.4 = 0.5.
Independence test: P(A)P(B) = 0.5 × 0.4 = 0.2, which equals P(A ∩ B), so the events are independent. (Equivalently P(A|B) = 0.5 = P(A).) Also P(A′ ∩ B′) = 1 − P(A ∪ B) = 0.3.
Traps that cost marks
- Multiplying P(A) by P(B) without first establishing independence.
- Treating "mutually exclusive" and "independent" as synonyms: exclusive events with positive probability are never independent.
- Dividing by the wrong denominator in conditional probability - the condition goes in the denominator.
- Reporting the variance when the standard deviation is asked, or forgetting |a| in σ_y = |a|σ_x.
- Mixing ordered and unordered counting in the same problem.
- Quoting r as evidence of causation, or as a measure of non-linear association.