Further Mathematics · Lesson 12

Conic sections: parabola, ellipse, hyperbola

WAECNECO

What the syllabus asks

WAEC: Coordinate geometry: conic sections: parabola, ellipse and hyperbola as loci (focus–directrix, eccentricity); standard equations, foci, directrices, vertices, axes, latus rectum, asymptotes; parametric form of the parabola; tangents

NECO: Coordinate geometry: conic sections: parabola, ellipse and hyperbola as loci (focus–directrix, eccentricity); standard equations, foci, directrices, vertices, axes, latus rectum, asymptotes; parametric form of the parabola; tangents

What you will learn

Describe the conics as sections of a cone and by focus–directrix and eccentricity.
Find the equation of a conic from its focus, directrix and eccentricity.
State and use properties of the parabola y² = 4ax and x² = 4ay, including parametric form and tangents.
State and use properties of the ellipse: foci, eccentricity, axes, directrices.
State and use properties of the hyperbola, including asymptotes and the rectangular hyperbola.
Reduce shifted equations to standard form and identify the conic.

Key points

Conics: PS = e × PM; e = 0 circle, e < 1 ellipse, e = 1 parabola, e > 1 hyperbola.
Parabola y² = 4ax: focus (a, 0), directrix x = −a, latus rectum 4a.
Parametric point (at², 2at); gradient 1/t; tangent ty = x + at².
Ellipse: c² = a² − b², foci (±c, 0), e = c/a, directrices x = ±a/e.
Hyperbola: c² = a² + b², e = c/a, asymptotes y = ±(b/a)x.
Rectangular hyperbola: x² − y² = a² or xy = c², e = √2.
Complete the square to find the vertex or centre of shifted conics.
Line meets conic: substitute; discriminant gives 2, 1 (tangent) or 0 points.

Start here: What conic sections are

Take a double cone, like two ice-cream cones joined at their tips. Slice it with a flat plane:

A slice parallel to the base gives a circle.
A slightly tilted slice gives a closed oval, an ellipse.
A slice parallel to the slanting side gives an open curve, a parabola.
A steep slice cutting both halves of the cone gives two separate curves, a hyperbola.

These curves are everywhere: the path of a thrown ball is (nearly) a parabola, planets move in ellipses around the sun, and the shadow of a lampshade on a wall can be a hyperbola.

For algebra, a more useful definition uses a fixed point (the focus S), a fixed line (the directrix) and a constant ratio (the eccentricity e).

A conic is the locus of a point P such that PS = e × PM, where PM is the perpendicular distance from P to the directrix.
e = 1: parabola · 0 < e < 1: ellipse · e > 1: hyperbola (a circle is the limiting case e = 0).

Worked example 12.1

Name the conic with eccentricity (a) 0.6; (b) 1; (c) 1.5.

(a) e < 1: ellipse.
(b) e = 1: parabola.
(c) e > 1: hyperbola.

Ellipse; parabola; hyperbola

⚠️ Common mistake: thinking a parabola is "half an ellipse". It is open and unbounded; an ellipse is closed.

Try it yourself. Which conic is a closed curve with two foci?

Show solution

The ellipse.

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Practice questions

1. Find the focus and directrix of y² = 28x.

Show worked answer

a = 7: (7, 0); x = −7.

2. Find the parabola with focus (−3, 0) and directrix x = 3.

Show worked answer

y² = −12x.

3. Find the tangent to y² = 12x at t = 1.

Show worked answer

a = 3; point (3, 6); y = x + 3.

4. For x²/169 + y²/144 = 1, find c and e.

Show worked answer

c² = 25, c = 5; e = 5/13.

5. For x²/36 − y²/64 = 1, find e and the asymptotes.

Show worked answer

c = 10; e = 5/3; y = ±(4/3)x.

Key words

Conic section: A curve formed by the intersection of a plane and a double cone.
Locus: The path traced by a point moving under a given condition.
Focus: A fixed point used to define a conic.
Directrix: A fixed line used to define a conic.
Eccentricity (e): The constant ratio PS/PM; it determines the type of conic.
Vertex: A point where a conic crosses its main axis.
Latus rectum: The chord through a focus perpendicular to the main axis.
Major axis: The longest diameter of an ellipse, length 2a.
Minor axis: The shortest diameter of an ellipse, length 2b.
Transverse axis: The axis of a hyperbola joining its vertices, length 2a.
Asymptote: A line that a curve approaches but never meets.
Parametric equations: Equations giving x and y in terms of a third variable (parameter).

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