The Open University since 2006
Alternatively you can skip the navigation by pressing 'Enter'.
The Great British Year: WinterSunday, 23rd October 2016 11:00 - EdenA frozen nation, but not a wasteland... Read more: The Great British Year: Winter
Inside The Commons: Reinventing The HouseAvailable until Saturday, 29th October 2016 19:00In the final part of this major four-part series, battles break out over the future of the House. Read more: Inside The Commons: Reinventing The House
The Bottom Line: Autumn 2016: Activist investorsAvailable for over a year
BBC Inside Science - 2016/2017 series: Lithium Batteries, HCFCs, Cell Mapping and Hunting DogsAvailable for over a year
Sleuths, Spies & Sorcerers: Andrew Marr’s Paperback Heroes: DetectivesAvailable until Friday, 18th November 2016 22:00
The Great British YearThe definitive portrait of the spectacular nature of the country over the course of one year. Read more: The Great British Year
Take the photographic memory testCan you capture scenes just by looking at them? Find out with our photographic memory test. Launch now: Take the photographic memory test
Liquidity managementIn this free course you will focus on liquidity management, one of the fundamental aspects... Try: Liquidity management now
Organisations and management accountingThis free course, Organisations and management accounting, examines the nature of organisations,... Try: Organisations and management accounting now
Attempts to answer problems in areas as diverse as science, technology and economics involve solving simultaneous linear equations. In this free course, Vectors and conics, we look at some of the equations that represent points, lines and planes in mathematics. We explore concepts such as Euclidean space, vectors, dot products and conics.
After studying this course, you should be able to:
- recognise the equation of a line in the plane
- determine the point of intersection of two lines in the plane, if it exists
- recognise the one-one correspondence between the set of points in three-dimensional space and the set of ordered triples of real numbers
- recognise the equation of a plane in three dimensions
- explain what are meant by a vector, a scalar multiple of a vector, and the sum and difference of two vectors.
- Learning outcomes
- 1 Coordinate geometry: points, planes and lines
- 1.1 Points, lines and distances in two-dimensional Euclidean space
- 1.2 Lines
- 1.3 Parallel and perpendicular lines
- 1.4 Intersection of two lines
- 1.5 Distance between two points in the plane
- 1.6 Points, planes, lines and distances in three-dimensional Euclidean space
- 1.7 Planes in three-dimensional Euclidean space
- 1.8 Intersection of two planes
- 1.9 Distance between points in three-dimensional Euclidean space
- 1.10 Further exercises
- 2 Vectors
- 3 Dot product
- 4 Conics
- Keep on learning
Study this free course
Enrol to access the full course, get recognition for the skills you learn, track your progress and on completion gain a statement of participation to demonstrate your learning to others. Make your learning visible!
4.6 Hyperbola (e > 1)
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
the focus F lies on the x-axis, and has coordinates (ae, 0), where a > 0;
the directrix d is the line with equation x = a/e.
Let P (x, y) be an arbitrary point on the hyperbola, and let M be the foot of the perpendicular from P to the directrix. Since FP = e × PM, by the definition of the hyperbola, it follows that FP2 = e2 × PM2; that is,
Multiplying out the brackets, we obtain
x2 − 2aex + a2e2 + y2 = e2x2 − 2aex + a2,
which simplifies to the equation
x2(e2 − 1) − y2 = a2(e2 − 1),
Substituting b for , so that b2 = a2(e2 − 1), we obtain the standard form of the equation of the hyperbola
This equation is symmetric in x and in y, so that the hyperbola also has a second focus F′ at (−ae, 0), and a second directrix d′ with equation x = −a/e.
The hyperbola intersects the x-axis at the points (±a, 0). We call the line segment joining the points (±a, 0) the major axis or transverse axis of the hyperbola, and the line segment joining the points (0, ±b) the minor axis or conjugate axis of the hyperbola (this is not a chord of the hyperbola). The origin is the centre of this hyperbola.
Each point with coordinates (a sec t, b tan t) lies on the hyperbola, since
Note: In general, sec2t = 1 + tan2t.
Then, just as for the parabola, we can check that
gives a parametric representation of the hyperbola.
Note: An alternative parametrisation, using hyperbolic functions, is x = acosht, y = bsinht (t ∈ ).
Two other features of the shape of the hyperbola stand out.
First, the hyperbola consists of two separate curves or branches.
Secondly, the lines with equations y = ±bx/a divide the plane into two pairs of opposite sectors; the branches of the hyperbola lie in one pair. As x → ±∞, the branches of the hyperbola get closer and closer to these two lines. We call the lines y = ±bx/a the asymptotes of the hyperbola.
We summarise these facts as follows:
Hyperbola in standard form
A hyperbola in standard form has equation
It can also be described by the parametric equations
It has foci (±ae, 0) and directrices x = ±a/e; its major axis is the line segment joining the points (±a, 0), and its minor axis is the line segment joining the points (0, ±b).
Let P be a point , t ∈ , on the hyperbola with equation x2 − 2y2 = 1.
(a) Determine the foci F and F′ of the hyperbola.
(b) Determine the gradients of FP and F′P, when these lines are not parallel to the y-axis.
(c) Find the point P on the hyperbola, in the first quadrant, for which FP is perpendicular to F′P.
(a) This hyperbola is of the form with a = 1 and , so . If e denotes the eccentricity of the hyperbola, so that b2 = a2(e2 − 1), we have
it follows that , so
In the general case, the foci are (±ae, 0); it follows that here the foci are .
(b) Let F and F′ be and , respectively. (It does not matter which way round these are chosen.)
Then the gradient of FP is
where we know that , since FP is not parallel to the y-axis.
Similarly, the gradient of F′P is
where we know that , since F′P is not parallel to the y-axis.
(c) When FP is perpendicular to F′P, we have
We may rewrite this in the form
so 2sec2t − 3 + tan2t = 0.
Since sec2t = 1 + tan2t, it follows that 3 tan2t = 1.
Since we are looking for a point P in the first quadrant, we choose .
When , we have . Since we are looking for a point P in the first quadrant, we choose .
It follows that the required point P has coordinates .
Copyright & revisions
Originally published: Wednesday, 18th May 2011
Last updated on: Tuesday, 15th March 2016
- Creative-Commons: The Open University is proud to release this free course under a Creative Commons licence. However, any third-party materials featured within it are used with permission and are not ours to give away. These materials are not subject to the Creative Commons licence. See terms and conditions. Full details can be found in the Acknowledgements and our FAQs section.
- This site has Copy Reuse Tracking enabled - see our FAQs for more information.
If you enjoyed this, why not follow a feed to find out when we have new things like it? Choose an RSS feed from the list below. (Don't know what to do with RSS feeds?)
Remember, you can also make your own, personal feed by combining tags from around OpenLearn.
All our alternative formats are free for you to download, for more information about the different formats we offer please see our FAQs. The most frequently used are Word (for accessibility), PDF (for print) and ePub and Kindle to download to eReaders*.
- Word (3 MB)
- PDF (5.1 MB)
- ePub 3.0 (24.5 MB)
- ePub 2.0 (2.2 MB)
- Kindle (1.8 MB)
- RSS (915 KB)
- HTML (24 MB)
- SCORM (24 MB)
- OUXML Package (49 KB)
- OUXML File (348 KB)
- IMS Common cartridge
*Please note you will need an ePub and Mobi reader for these formats.