Patterns occur everywhere in art, nature, science and especially mathematics. Being able to recognise, describe and use these patterns is an important skill that helps you to tackle a wide variety of different problems. This unit explores some of these patterns ranging from ancient number patterns to the latest mathematical research. It also looks at some useful practical applications. You will see how to describe some patterns mathematically as formulas and how these can be used to solve pro
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## Audio material

The audio extracts are taken from M208 Pure Mathematics. Â© 2006
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## Exercise 58

Determine the equation of the circle with centre (2, 1) and radius 3.

The equation of the circle is

(
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You have already met the parabola, ellipse and hyperbola. So far, you have considered the equation of a conic only when it is in standard form; that is, when the centre of the conic (if it has a centre) is at the origin, and the axes of the conic are parallel to the x- and y-axes. However, most of the conics that arise in calculations are not in standard form.

We have seen that any circle can be described by an equation of the form

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If the eccentricity e of a hyperbola is equal to âˆš2, then e2Â =Â 2 and bÂ =Â a. Then the asymptotes of the hyperbola have equations yÂ =Â Â±x, so they are at right angles. A hyperbola whose asymptotes are at right angles is called a rectangular hyperbola.

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A hyperbola is the set of points P in the plane whose distances from a fixed point F are e times their distances from a fixed line d, where e > 1. We obtain a hyperbola in standard form if

1. the focus F lies on the x-axis, and has coordinates (ae, 0), where a > 0;

2. the directrix d is the line with equation xÂ =Â a/e.

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An ellipse with eccentricity e (where 0 < e < 1) is the set of points P in the plane whose distances from a fixed point F are e times their distances from a fixed line d. We obtain such an ellipse in standard form if

1. the focus F lies on the x-axis, and has coordinates (ae, 0), where a > 0;

2. the directrix d is the line with equation xÂ =Â a
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4.4 Parabola (e = 1)

A parabola is defined to be the set of points P in the plane whose distances from a fixed point F are equal to their distances from a fixed line d. We obtain a parabola in standard form if

1. the focus F lies on the x-axis, and has coordinates (a, 0), where a > 0;

2. the directrix d is the line with equation xÂ =Â âˆ’a.

Thus the origin lies on t
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4.3 Focus–directrix definitions of the non-degenerate conics

Earlier, we defined the conic sections as the curves of intersection of planes with a double cone. One of these conic sections, the circle, can be defined as the set of points a fixed distance from a fixed point.

Here we define the other non-degenerate conics, the parabola, ellipse and hyperbola, as sets of points that satisfy a somewhat similar condition.

These three non-degenerate conics (the parabola, ellipse and hyperbola) can be defined as the set of points P in
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4.2 Circles

Recall that a circle in 2 is the set of points (x, y) that lie at a fixed distance, called the radius, from a fixed point, called the centre of the circle. We can use the techniques of coordinate geometry to find the equation of a circle with a given centre and radius.
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4.1 Conic sections

Conic section is the collective name given to the shapes that we obtain by taking different plane slices through a double cone. The shapes that we obtain from these cross-sections are drawn below. It is thought that the Greek mathematician Menaechmus discovered the conic sections around 350 bc.

3.4 Further exercises

## Exercise 47

• (a) Find the angle between each of the pairs of vectors:

(3, 1) and (1, âˆ’2); i + 2j and âˆ’3i + j âˆ’ 2k
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3.3 Equation of a plane in three-dimensional Euclidean space

We stated in Section 1.7 that the general form of the equation of a plane in 3 is

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3.2 Post-audio exercises

## Exercise 42

Let u and v be the position vectors (6, 8) and (âˆ’12, 5), respectively.

• (a) Sketch u and v on a single diagram. On th
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3.1 Definition, properties and some applications

In the previous section we saw how to add two vectors and how to multiply a vector by a scalar, but we did not consider how to multiply two vectors. There are two different ways in which we can multiply two vectors, known as the dot product (or scalar product) and the vector product. They are given these names because the result of the first is a scalar and the result of the second is a vector. (We shall not consider vector products in this course.)

In the audio sec
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2.7 Further exercises

## Exercise 31

Let pÂ =Â 2i âˆ’ 3j + k and qÂ =Â âˆ’i âˆ’2j âˆ’4k be two vectors in Author(s): The Open University

2.4 Components and the arithmetic of vectors

We introduce now a different method of representing vectors, which will make the manipulation of vectors much easier. Thus we shall avoid having to solve problems involving vectors by drawing the vectors and making measurements, which is very time-consuming and never very accurate.

We can think of a vector as a translation, that is, as representing a movement by a certain amount in a given direction. Then we can use the Cartesian axes in the plane or in Author(s): The Open University

We saw above that the vector 2v can be regarded as the vector v â€˜followed byâ€™ the vector v; we can also quite naturally describe this vector as being the â€˜sumâ€™, v + v, of the vector with itself.

Analogously, if p is the vector 2 cm E and q is the vector 3 cm NE, we can think of the â€˜sumâ€™ p + q of the vectors as follows. Starting from a given point, O say, draw the vector p; starting from its finishi
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2.2 Multiplication by a scalar

In the collection of vectors sketched in Section 2.1, although v is not equal to c, the vectors v and c are closely related: c is a vector in the same direction as v, but it is twice as long as v. Thus it is natural to write 2v for c, since we can think of a journey repre
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2.1 Definitions

In this section we introduce an alternative way of describing points in the plane 2 or in three-dimensional space 3;
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