Geometry & Algebra

1. Analytic Geometry — Lines and Circles

• Line: slope-intercept form y = mx + b (m = slope, b = y-intercept); point-slope form y − y₁ = m(x − x₁).
• Circle: (x − a)² + (y − b)² = r², centered at (a, b) with radius r.

Common mistakes: in the circle equation, mixing up the sign of the center coordinates — the center is (a, b), NOT (−a, −b), even though the equation has (x−a) and (y−b).

2. Conic Sections — Ellipse, Hyperbola, Parabola

• Ellipse (centered at origin): x²/a² + y²/b² = 1. If a > b, the major axis is along the x-axis with length 2a.
  • Eccentricity: e = c/a, where c² = a² − b² (a > b case). 0 < e < 1 for an ellipse.
• Hyperbola: x²/a² − y²/b² = 1. Eccentricity e = c/a where c² = a² + b², and e > 1.
• Parabola: y² = 4px (opens along the x-axis) or x² = 4py (opens along the y-axis), where p is the distance from vertex to focus.

Common mistakes: using c² = a² − b² for a hyperbola (it should be c² = a² + b² — the relationship is different from the ellipse); forgetting eccentricity is always computed as c/a, not c/b.

3. Vectors

• A vector has both magnitude and direction. In 2D: v = (x, y), magnitude |v| = √(x² + y²).
• Addition/subtraction: component-wise, (x₁,y₁) + (x₂,y₂) = (x₁+x₂, y₁+y₂).
• Scalar multiplication: k·(x,y) = (kx, ky).
• Dot product: a·b = x₁x₂ + y₁y₂ = |a||b|cos(θ), where θ is the angle between them. If a·b = 0, the vectors are perpendicular.

Common mistakes: confusing the dot product (a scalar result) with vector addition (a vector result); forgetting the dot product formula uses cos(θ), not sin(θ).

4. Complex Numbers

• A complex number has the form a + bi, where i² = −1.
• Addition: (a+bi) + (c+di) = (a+c) + (b+d)i.
• Multiplication: (a+bi)(c+di) = (ac − bd) + (ad + bc)i (expand and replace i² with −1).
• Modulus (magnitude): |a+bi| = √(a² + b²).

Common mistakes: forgetting to replace i² with −1 during multiplication, leading to a sign error; confusing the modulus formula with the Pythagorean-looking vector magnitude (they are structurally the same, but students sometimes mix up a and b when reading the problem).

5. Solid Geometry

• 3D distance formula: between (x₁,y₁,z₁) and (x₂,y₂,z₂): d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²].
• Space diagonal of a rectangular box with sides a, b, c: d = √(a² + b² + c²).
• Volumes: rectangular prism = length×width×height; cylinder = πr²h; sphere = (4/3)πr³; cone = (1/3)πr²h; pyramid = (1/3)×base area×height.

Common mistakes: using the 2D distance formula (missing the z-term) for 3D problems; forgetting the 1/3 factor for cones and pyramids.