The Open University since 2006
Alternatively you can skip the navigation by pressing 'Enter'.

TV, Radio & Events menu item
Coming up
Thinking Allowed 2016: The Flaneur  Walking in the City
Monday, 2nd May 2016 00:15  BBC Radio 4In this themed programme Laurie Taylor and guests look at the meaning of the urban stroller, past and present. Read more: Thinking Allowed 2016: The Flaneur  Walking in the CityLife In Cold Blood: Dragons Of The Dry
Monday, 2nd May 2016 09:00  YesterdayLife In Cold Blood: Dragons Of The Dry
Monday, 2nd May 2016 14:00  YesterdayLife In Cold Blood: Dragons Of The Dry
Monday, 2nd May 2016 19:00  YesterdayCatch up
Shakespeare Speaks: A pound of flesh
Available for over a yearKing James needs to stay awake during this performance of The Merchant of Venice to find out why Shylock is talking... Read more: Shakespeare Speaks: A pound of fleshEveryday Miracles: The Genius of Sofas, Stockings & Scanners: Away
Available until Saturday, 28th May 2016 21:00Thinking Allowed 2016: The Flaneur  Walking in the City
Available for over a yearAll in the Mind  Summer 2016: All in the Mind Awards and psychology in films
Available for over a year 
Get Started menu item
Welcome to OpenLearn
Get the most out of OpenLearn  Sign in
OpenLearn is the home of free learning from the UK's largest university, The Open University. We've got thousands of ways to learn  including 800 free courses you can dip into. More about this site
Inspire me
Life In Cold Blood
David Attenborough explores the lives of amphibians and reptiles  cold blood, but amazing... Watch now: Life In Cold BloodTake the photographic memory test
Can you capture scenes just by looking at them? Find out with our photographic memory test. Launch now: Take the photographic memory testFree courses
Technology, innovation and management
Develop a critical understanding of technological innovation and management. This free course,... Try: Technology, innovation and management nowForensic psychology
In this free course, Forensic psychology, you will discover how psychology can help obtain... Try: Forensic psychology nowBecome an OU student
Learning to learn
Try our free course and get the most out of all our free courses Try: Learning to learn now
 You are here:
 Home
 Science, Maths & Technology
 Mathematics and Statistics
 Vectors and conics
 4.6 Hyperbola (e > 1)
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 oneone correspondence between the set of points in threedimensional 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.
 Duration 20 hours
 Updated Tuesday 15th March 2016
 Intermediate level
 Posted under Mathematics and Statistics
Contents
 Introduction
 Learning outcomes
 1 Coordinate geometry: points, planes and lines
 1.1 Points, lines and distances in twodimensional 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 threedimensional Euclidean space
 1.7 Planes in threedimensional Euclidean space
 1.8 Intersection of two planes
 1.9 Distance between points in threedimensional Euclidean space
 1.10 Further exercises
 2 Vectors
 3 Dot product
 4 Conics
 Conclusion
 Keep on learning
 Acknowledgements
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 xaxis, 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 FP^{2} = e^{2} × PM^{2}; that is,
Multiplying out the brackets, we obtain
x^{2} − 2aex + a_{2}e_{2} + y_{2} = e_{2}x_{2} − 2aex + a_{2},
which simplifies to the equation
x_{2}(e_{2} − 1) − y_{2} = a_{2}(e_{2} − 1),
that is,
Substituting b for , so that b^{2} = a^{2}(e^{2} − 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 xaxis 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, sec^{2}t = 1 + tan^{2}t.
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).
Example 57
Let P be a point , t ∈ , on the hyperbola with equation x^{2} − 2y^{2} = 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 yaxis.
(c) Find the point P on the hyperbola, in the first quadrant, for which FP is perpendicular to F′P.
Answer
(a) This hyperbola is of the form with a = 1 and , so . If e denotes the eccentricity of the hyperbola, so that b^{2} = a^{2}(e^{2} − 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 yaxis.
Similarly, the gradient of F′P is
where we know that , since F′P is not parallel to the yaxis.
(c) When FP is perpendicular to F′P, we have
We may rewrite this in the form
so 2sec^{2}t − 3 + tan^{2}t = 0.
Since sec^{2}t = 1 + tan^{2}t, it follows that 3 tan^{2}t = 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 .
Comments
We invite you to discuss this subject, but remember this is a public forum.
Please be polite, and avoid your passions turning into
contempt for others. We may delete posts that are rude or aggressive; or edit posts containing contact details or links to other websites.
Be the first to post a comment
Copyright & revisions
Publication details

Originally published: Wednesday, 18th May 2011

Last updated on: Tuesday, 15th March 2016
Copyright information
 CreativeCommons: The Open University is proud to release this free course under a Creative Commons licence. However, any thirdparty 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.
Feeds
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.
Alternative formats
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.
Tags, Ratings and Social Bookmarking
About this site
OpenLearn: free learning from The Open University