1 Introduction to Conic Sections
Conic sections are curves formed by the intersection of a plane and a double cone. They are defined as the locus of points satisfying a geometric condition.
| Conic | Locus Definition | Standard Equation |
|---|---|---|
| Circle | All points equidistant from a centre | (x−h)² + (y−k)² = r² |
| Ellipse | All points where sum of distances from two foci is constant | x²/a² + y²/b² = 1 |
| Parabola | All points equidistant from a focus and a directrix line | (x−h)² = 4c(y−k) |
| Hyperbola | All points where difference of distances from two foci is constant | x²/a² − y²/b² = 1 |
2 The Circle
Standard form
(x − h)² + (y − k)² = r²
Centre
(h, k)
Radius
r
General form
x² + y² + Dx + Ey + F = 0 (expand and collect terms)
Standard ↔ General Form
- Standard → General: Expand (x−h)² and (y−k)², combine constants.
- General → Standard: Complete the square for both x and y, then read off h, k, r.
✏️
Example (General → Standard): x² + y² − 4x − 6y + 4 = 0
(x² − 4x + 4) + (y² − 6y + 9) = −4 + 4 + 9
(x − 2)² + (y − 3)² = 9 → centre (2, 3), radius 3
(x² − 4x + 4) + (y² − 6y + 9) = −4 + 4 + 9
(x − 2)² + (y − 3)² = 9 → centre (2, 3), radius 3
3 The Ellipse
An ellipse is a "stretched circle" with two focal points (foci). The sum of distances from any point on the ellipse to both foci is constant = 2a (the length of the major axis).
Standard Form — Horizontal Ellipse (wider side along x-axis)
Equation
x²/a² + y²/b² = 1 (a > b)
Major axis length
2a (along x-axis)
Minor axis length
2b (along y-axis)
Foci location
(±c, 0) where c² = a² − b² (or b² = a² − c²)
Latus rectum
Line through a focus perpendicular to major axis
Standard Form — Vertical Ellipse (taller side along y-axis)
Equation
x²/a² + y²/b² = 1 (b > a)
Major axis
Along y-axis, length 2b
Foci
(0, ±c) where c² = b² − a²
💡
Foci are always on the major axis (the longer one). The closer the foci to the centre, the more circular the ellipse.
⚠️
Key relation for ellipse: a² = b² + c² (largest value = other two combined). For hyperbola it's different!
4 The Conic Parabola
As a conic, a parabola is defined by a focus (a point) and a directrix (a line). Every point on the parabola is equidistant from both.
Four Possible Orientations
| Opens | Standard Form | Vertex | Focus | Directrix |
|---|---|---|---|---|
| Up | (x−h)² = 4c(y−k) (c > 0) | (h, k) | (h, k+c) | y = k−c |
| Down | (x−h)² = −4c(y−k) (c > 0) | (h, k) | (h, k−c) | y = k+c |
| Right | (y−k)² = 4c(x−h) (c > 0) | (h, k) | (h+c, k) | x = h−c |
| Left | (y−k)² = −4c(x−h) (c > 0) | (h, k) | (h−c, k) | x = h+c |
📐
Latus Rectum = 4c — a chord through the focus, parallel to the directrix. Use it as a graphing aid.
Relation to Quadratic
Quadratic form
y = ax² + bx + c (vertex form: y = a(x−h)² + k)
Conic parabola form
(x−h)² = 4c(y−k)
Connection
a = 1/(4c) → c = 1/(4a)
Completing the Square for Conic Parabola
To go from general form to standard form:
- Identify which variable has the quadratic term (x² or y²).
- Move all other terms to the right side.
- Complete the square: add (b/2)² to both sides.
- Factor the perfect square trinomial.
- Read off h, k, and c.
5 The Hyperbola (Centred at Origin)
A hyperbola has two branches that open in opposite directions. Unlike the ellipse, the focal relation uses a difference of distances.
Horizontal Hyperbola (opens left/right)
Equation
x²/a² − y²/b² = 1
Vertices
(±a, 0) — on the x-axis
Asymptotes
y = ±(b/a)x
Foci
(±c, 0) where c² = a² + b²
Vertical Hyperbola (opens up/down)
Equation
y²/b² − x²/a² = 1 (or −x²/a² + y²/b² = 1)
Vertices
(0, ±b) — on the y-axis
Asymptotes
y = ±(b/a)x
Foci
(0, ±c) where c² = a² + b²
⚠️
Ellipse vs Hyperbola: Ellipse: a² = b² + c² (subtract). Hyperbola: c² = a² + b² (add). The hyperbola's c is LARGER than both a and b.
Graphing a Hyperbola — 5 Steps
- Identify a, b, and whether it opens horizontally or vertically.
- Draw the central box: width = 2a, height = 2b.
- Draw the asymptotes through the corners of the box.
- Plot the foci at c = √(a² + b²) from the centre.
- Sketch the two branches touching the vertices, approaching the asymptotes.
6 Completing the Square — General Method
Used to convert any conic from general form to standard form.
- Group x terms and y terms on the left side.
- Move the constant to the right side.
- For each variable group: add (b/2)² to both sides (where b is the coefficient of the linear term).
- Factor each group as a perfect square.
- Identify the conic type and parameters.
✏️
Example: y² + 6y − 4x − 19 = 0
y² + 6y = 4x + 19
y² + 6y + 9 = 4x + 19 + 9 [add (6/2)² = 9]
(y + 3)² = 4x + 28 = 4(x + 7)
→ Horizontal parabola, vertex (−7, −3), c = 1
y² + 6y = 4x + 19
y² + 6y + 9 = 4x + 19 + 9 [add (6/2)² = 9]
(y + 3)² = 4x + 28 = 4(x + 7)
→ Horizontal parabola, vertex (−7, −3), c = 1
7 Common Mistakes
| Mistake | Fix |
|---|---|
| Confusing ellipse and hyperbola formulas | Ellipse: + between fractions, sum = 1. Hyperbola: − between fractions. The sign changes everything. |
| Wrong focal formula | Ellipse: c² = a² − b² · Hyperbola: c² = a² + b². Learn which one adds and which one subtracts. |
| Forgetting to add to BOTH sides when completing square | Adding (b/2)² to the left must be balanced by adding the same to the right. |
| Reading h and k with wrong signs | (x − h)² means h is positive even if you see a minus. (x + 3)² → h = −3. |
| Conic parabola: confusing c with the quadratic a | a in y = ax² equals 1/(4c). They are related but not the same number. |