Lesson 12 of 12 · 15 min

Chapter review

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Must-know facts

18 facts

1β = 90°: circle; α < β < 90°: ellipse; β = α: parabola; 0 ≤ β < α: hyperbola.
2Plane through the vertex: a point, a line (degenerate parabola) or a pair of lines (degenerate hyperbola).
3Circle with centre (h, k), radius r: (x − h)² + (y − k)² = r².
4x² + y² + 8x + 10y − 8 = 0 has centre (−4, −5) and radius 7.
5Parabola: distance from focus = distance from directrix.
6y² = 4ax: focus (a, 0), directrix x = −a, axis the x-axis, latus rectum 4a.
7y² = 8x: focus (2, 0), directrix x = −2, latus rectum 8.
8Ellipse: PF₁ + PF₂ = 2a, with a² = b² + c² and e = c/a < 1.
9x²/a² + y²/b² = 1 (a > b): foci (±c, 0), vertices (±a, 0).
10The larger denominator decides the major axis of an ellipse.
11x²/25 + y²/9 = 1: foci (±4, 0), e = 4/5, latus rectum 18/5.
12Latus rectum of an ellipse or a hyperbola: 2b²/a.
13Hyperbola: |PF₁ − PF₂| = 2a, with c² = a² + b² and e = c/a > 1.
14The positive term decides the transverse axis of a hyperbola.
15x²/9 − y²/16 = 1: foci (±5, 0), vertices (±3, 0), e = 5/3, latus rectum 32/3.
16a = b gives an equilateral hyperbola.
17Parabolic mirror, focus 5 cm, depth 45 cm: rim 60 cm across.
18Rod of 15 cm with AP = 6 cm traces x²/81 + y²/36 = 1.

Common traps

Where marks are lost

Using c² = a² − b² for a hyperbola.

Ellipse: c² = a² − b². Hyperbola: c² = a² + b². The foci of a hyperbola are farther out than its vertices.

Taking a from the x² term every time.

In an ellipse a goes with the larger denominator; in a hyperbola a goes with the positive term, whichever variable that is.

Reading the focus of x² = 12y as (3, 0).

An x² term means the axis is the y-axis, so the focus is (0, 3) and the directrix y = −3.

Giving the latus rectum of y² = 4ax as 2a.

2a is only half of it; the full chord through the focus is 4a.

Forgetting to divide through to get 1 on the right: treating 9x² + 4y² = 36 as a² = 9, b² = 4.

Divide by 36 first: x²/4 + y²/9 = 1, so a = 3 lies along the y-axis.

Quoting an eccentricity above 1 for an ellipse or below 1 for a hyperbola.

Ellipse: e = c/a < 1 since c < a. Hyperbola: e = c/a > 1 since c > a. A value on the wrong side means a wrong relation was used.

Completing the square with the wrong constant on the right.

Whatever is added inside the brackets is added to the right side too: 8 + 16 + 25 = 49.

Keeping a negative root for a length, as in a² + 18a − 144 = 0.

a is a length, so a = 6 and the root −24 is rejected.

Formulas

10 to know

Circle

(x − h)² + (y − k)² = r²

Centre (h, k), radius r; x² + y² = r² when the centre is the origin.

Parabola (opens right)

y² = 4ax

Focus (a, 0), directrix x = −a, latus rectum 4a.

Other parabolas

y² = −4ax, x² = 4ay, x² = −4ay

Open left, up, down; a > 0.

Ellipse (major axis on x)

x²/a² + y²/b² = 1, a > b

Foci (±c, 0), vertices (±a, 0).

Ellipse (major axis on y)

x²/b² + y²/a² = 1, a > b

Foci (0, ±c), vertices (0, ±a).

Ellipse relation

c² = a² − b², e = c/a < 1

Foci are ae from the centre.

Hyperbola (transverse on x)

x²/a² − y²/b² = 1

Foci (±c, 0), vertices (±a, 0).

Hyperbola (transverse on y)

y²/a² − x²/b² = 1

Foci (0, ±c), vertices (0, ±a).

Hyperbola relation

c² = a² + b², e = c/a > 1

a = b gives an equilateral hyperbola.

Latus rectum

Parabola 4a; ellipse and hyperbola 2b²/a

Key terms

14 terms

Double napped cone: The surface swept by a line turning about a fixed axis it crosses at a fixed angle; two cones joined at the vertex.
Generator: The turning line whose positions make up the cone.
Nappe: Either of the two halves of the cone on each side of the vertex.
Conic section: The curve in which a plane cuts a double cone.
Degenerate conic: A point, a line or a pair of lines, got when the plane passes through the vertex.
Focus: A fixed point used in the distance definition of a parabola, ellipse or hyperbola.
Directrix: The fixed line of a parabola; every point of the curve is as far from it as from the focus.
Vertex: A point where the curve crosses its axis (major or transverse axis for an ellipse or hyperbola).
Latus rectum: The chord through a focus perpendicular to the axis (major or transverse axis), ending on the curve.
Major and minor axes: For an ellipse, the chord through the foci (length 2a) and the chord perpendicular to it through the centre (length 2b).
Transverse and conjugate axes: For a hyperbola, the line through the foci and the line through the centre perpendicular to it; lengths 2a and 2b.
Eccentricity: e = c/a, centre-to-focus over centre-to-vertex; below 1 for an ellipse, above 1 for a hyperbola.
Equilateral hyperbola: A hyperbola with a = b.
Locus: The path traced by a point that moves under a given condition, such as the ellipse traced by a point of a sliding rod.