A7LAM CSCA ACADEMYCSCA Academy / Mathematics / Geometry and algebra
CSCA Mathematics · Lesson 11 of 16Ellipses, hyperbolas and parabolas
Ellipses, hyperboles et paraboles · القطع الناقص والزائد والمكافئ
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Conic sections share one idea: a point moves following a rule about its distances to a focus. Learn the standard equations and how a, b and c are related.
Key ideas
- Ellipse x²/a² + y²/b² = 1 (a > b > 0): c² = a² − b², foci (±c, 0).
- Hyperbola x²/a² − y²/b² = 1: c² = a² + b², asymptotes y = ±(b/a)x.
- Parabola y² = 2px (p > 0): focus (p/2, 0), directrix x = −p/2.
- Eccentricity e = c/a: e < 1 ellipse, e = 1 parabola, e > 1 hyperbola.
Formulas to know
- Ellipse: PF₁ + PF₂ = 2a
- Hyperbola: |PF₁ − PF₂| = 2a
- Parabola: distance to focus = distance to directrix
Worked example
Find the foci of the ellipse x²/25 + y²/9 = 1.
- Read: a² = 25, b² = 9
- Ellipse: c² = a² − b² = 16, so c = 4
- Foci: on the x-axis at (±c, 0)
Answer: (−4, 0) and (4, 0)
Practice questions
1. Focus of the parabola y² = 8x?
(2, 0) ✓ · (4, 0) · (8, 0) · (0, 2)
2p = 8, p = 4, focus (p/2, 0) = (2, 0).
2. Eccentricity of x²/25 + y²/9 = 1?
3/5 · 4/5 ✓ · 5/4 · 3/4
c = 4, a = 5, e = 4/5.
3. Asymptotes of x²/4 − y²/9 = 1?
y = ±(2/3)x · y = ±(3/2)x ✓ · y = ±(9/4)x · y = ±3x
a = 2, b = 3, y = ±(b/a)x.