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.

Author(s): The Open University

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.

Author(s): The Open University

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
Author(s): The Open University

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

2.3 Addition of vectors

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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1.10 Further exercises

## Exercise 12

Determine the equation of the line through each of the following pairs of points. Show that both equations can be written in the form ax + byÂ =Â c, for some real numbers a
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1.9 Distance between points in three-dimensional Euclidean space

You saw in Section 1.5 that the distance between two points (x1, y1) and (x2, y2) in the plane is given by Author(s): The Open University

1.8 Intersection of two planes

We saw earlier that two arbitrary lines in 2 may intersect, be parallel, or coincide. In an analogous way, two arbitrary planes in
Author(s): The Open University

1.7 Planes in three-dimensional Euclidean space

We now look at the general form of the equation of a plane in 3.

Three planes whose equations are easy to find are those that contain a pair of axes. For example, the (x, y)-plane is the plane that contains the x-axis and the y-axis. Points which lie in this plane are prec
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