Conic sections are curves formed by the intersection of a plane with a double-napped cone. Depending on the angle of the plane, we get four types of curves: circle, parabola, ellipse, and hyperbola. These curves appear throughout science and engineering, from the orbits of planets (ellipses) to the shape of satellite dishes (parabolas).
- Circle: The set of all points in a plane equidistant from a fixed point (the centre).
- Standard equation: (x - h)2 + (y - k)2 = r2, centre (h, k), radius r
- Simplified (centre at origin): x2 + y2 = r2
- Parabola: The set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix).
- Opens right: y2 = 4ax (focus at (a, 0), directrix x = -a)
- Opens left: y2 = -4ax
- Opens up: x2 = 4ay
- Opens down: x2 = -4ay
- The axis of the parabola passes through the focus and is perpendicular to the directrix.
- The vertex is midway between the focus and directrix.
- Ellipse: The set of all points such that the sum of distances from two fixed points (the foci) is constant.
- Standard form (major axis along x): x2/a2 + y2/b2 = 1, where a > b > 0
- c2 = a2 - b2, eccentricity e = c/a (0 < e < 1)
- Foci at (±c, 0), vertices at (±a, 0)
- Hyperbola: The set of all points such that the difference of distances from two fixed points is constant.
- Standard form: x2/a2 - y2/b2 = 1
- c2 = a2 + b2, eccentricity e = c/a (e > 1)
- Asymptotes: y = ±(b/a)x
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Worked Examples
Find the equation of the circle with centre (3, -2) and radius 5.
(x - 3)2 + (y + 2)2 = 25
Find the centre and radius of x2 + y2 - 6x + 4y + 4 = 0.
Complete the square: (x - 3)2 - 9 + (y + 2)2 - 4 + 4 = 0
(x - 3)2 + (y + 2)2 = 9. Centre (3, -2), radius 3.
For the parabola y2 = 12x, find focus, directrix, and latus rectum.
Comparing y2 = 4ax: 4a = 12, a = 3. Focus (3, 0), directrix x = -3, latus rectum length = 4a = 12.
Find the equation of ellipse with foci (±3, 0) and a = 5.
c = 3, a = 5. b2 = a2 - c2 = 25 - 9 = 16. Equation: x2/25 + y2/16 = 1
Find the eccentricity of the ellipse x2/16 + y2/9 = 1.
a2 = 16, b2 = 9. c2 = 16 - 9 = 7. e = c/a = √(7)/4.
Find the foci and eccentricity of the hyperbola x2/9 - y2/16 = 1.
a2 = 9, b2 = 16. c2 = 9 + 16 = 25, c = 5. Foci (±5, 0), e = 5/3.
Find the length of the latus rectum of the ellipse x2/25 + y2/16 = 1.
Length of latus rectum = 2b2/a = 2(16)/5 = 32/5
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Common mistakes
- For an ellipse, a > b; the larger denominator gives a2. Do not confuse a and b.
- For a hyperbola, c2 = a2 + b2 (note: PLUS sign), unlike the ellipse where c2 = a2 - b2.
- The latus rectum of a parabola y2 = 4ax has length 4a, not 2a or a.
Summary
Know the standard equations and properties (focus, directrix, eccentricity, latus rectum) for all four conics. The relationship between a, b, and c is the most frequently tested formula.