Overview
This chapter covers the conic sections — circle, parabola, ellipse and hyperbola — and their standard equations, foci, directrices and eccentricities.
1. Circle
Standard: (x − h)² + (y − k)² = r²
General : x² + y² + 2gx + 2fy + c = 0
Centre (−g, −f), radius √(g² + f² − c)
2. Parabola (vertex at origin)
y² = 4ax : focus (a, 0), directrix x = −a, latus rectum 4a
x² = 4ay : focus (0, a), directrix y = −a
3. Ellipse
x²/a² + y²/b² = 1 (a > b)
Foci (±c, 0) with c² = a² − b² ; eccentricity e = c/a < 1
Length of major axis = 2a, minor axis = 2b
4. Hyperbola
x²/a² − y²/b² = 1
Foci (±c, 0) with c² = a² + b² ; eccentricity e = c/a > 1
Asymptotes: y = ±(b/a)x
Worked Examples
Example 1 — Circle from general form
Find the centre and radius of x² + y² − 4x + 6y − 12 = 0.
2g = −4 → g = −2 ; 2f = 6 → f = 3 ; c = −12
Centre (−g, −f) = (2, −3)
radius = √(g² + f² − c) = √(4 + 9 + 12) = √25 = 5
Example 2 — Parabola
Find focus and directrix of y² = 12x.
4a = 12 → a = 3
Focus (3, 0), directrix x = −3, latus rectum = 12
Example 3 — Ellipse eccentricity
For x²/25 + y²/9 = 1, find foci and eccentricity.
a² = 25, b² = 9 → c² = 25 − 9 = 16 → c = 4
e = c/a = 4/5 ; foci (±4, 0)
Example 4 — Hyperbola
For x²/16 − y²/9 = 1, find eccentricity and asymptotes.
a² = 16, b² = 9 → c² = 16 + 9 = 25 → c = 5
e = 5/4 ; asymptotes y = ±(3/4)x
Common Mistakes
- Using c² = a² − b² for a hyperbola (it is a² + b²).
- Forgetting that a > b for the standard horizontal ellipse.
- Reading the centre as (g, f) instead of (−g, −f).
- Confusing latus rectum length with 2a.
Exam Tips & Revision Checklist
- Identify the conic from its equation first, then apply its formula set.
- Remember the eccentricity values: circle 0, ellipse <1, parabola 1, hyperbola >1.
- Practised: circle centre/radius, parabola focus, ellipse foci/e, hyperbola e/asymptotes.
Frequently Asked Questions
(x − h)² + (y − k)² = r², centre (h, k), radius r. The general form is x² + y² + 2gx + 2fy + c = 0 with centre (−g, −f) and radius √(g² + f² − c).