1. Home
  2. College algebra
  3. Analytic geometry
  4. The ellipse

8.1 The ellipse  (Page 3/16)

<< Chapter < Page
  College algebra     Page 3 / 16
Chapter >> Page >

Standard forms of the equation of an ellipse with center (0,0)

The standard form of the equation of an ellipse with center   ( 0 , 0 )   and major axis on the x-axis is

x 2 a 2 + y 2 b 2 = 1

where

  • a > b
  • the length of the major axis is   2 a
  • the coordinates of the vertices are   ( ± a , 0 )
  • the length of the minor axis is   2 b
  • the coordinates of the co-vertices are   ( 0, ± b )
  • the coordinates of the foci are   ( ± c , 0 ) , where   c 2 = a 2 − b 2 .   See [link] a

The standard form of the equation of an ellipse with center   ( 0 , 0 )   and major axis on the y-axis is

x 2 b 2 + y 2 a 2 = 1

where

  • a > b
  • the length of the major axis is   2 a
  • the coordinates of the vertices are   ( 0, ± a )
  • the length of the minor axis is   2 b
  • the coordinates of the co-vertices are   ( ± b , 0 )
  • the coordinates of the foci are   ( 0, ± c ) , where   c 2 = a 2 − b 2 .   See [link] b

Note that the vertices, co-vertices, and foci are related by the equation   c 2 = a 2 − b 2 .   When we are given the coordinates of the foci and vertices of an ellipse, we can use this relationship to find the equation of the ellipse in standard form.

(a) Horizontal ellipse with center   ( 0 , 0 )   (b) Vertical ellipse with center   ( 0 , 0 )

Given the vertices and foci of an ellipse centered at the origin, write its equation in standard form.

  1. Determine whether the major axis lies on the x - or y -axis.
    1. If the given coordinates of the vertices and foci have the form   ( ± a , 0 )   and   ( ± c , 0 )   respectively, then the major axis is the x -axis. Use the standard form   x 2 a 2 + y 2 b 2 = 1.
    2. If the given coordinates of the vertices and foci have the form   ( 0, ± a )   and   ( ± c , 0 ) , respectively, then the major axis is the y -axis. Use the standard form   x 2 b 2 + y 2 a 2 = 1.
  2. Use the equation   c 2 = a 2 − b 2 ,   along with the given coordinates of the vertices and foci, to solve for   b 2 .
  3. Substitute the values for   a 2   and   b 2   into the standard form of the equation determined in Step 1.

Writing the equation of an ellipse centered at the origin in standard form

What is the standard form equation of the ellipse that has vertices   ( ± 8 , 0 )   and foci   ( ± 5 , 0 ) ?  

The foci are on the x -axis, so the major axis is the x -axis. Thus, the equation will have the form

x 2 a 2 + y 2 b 2 = 1

The vertices are   ( ± 8 , 0 ) , so   a = 8   and   a 2 = 64.

The foci are   ( ± 5 , 0 ) , so   c = 5   and   c 2 = 25.

We know that the vertices and foci are related by the equation   c 2 = a 2 − b 2 .   Solving for   b 2 , we have:

c 2 = a 2 − b 2 25 = 64 − b 2 Substitute for  c 2  and  a 2 . b 2 = 39 Solve for  b 2 .

Now we need only substitute   a 2 = 64   and   b 2 = 39   into the standard form of the equation. The equation of the ellipse is   x 2 64 + y 2 39 = 1.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

What is the standard form equation of the ellipse that has vertices   ( 0, ± 4 )   and foci   ( 0, ± 15 ) ?

x 2 + y 2 16 = 1

Got questions? Get instant answers now!

Can we write the equation of an ellipse centered at the origin given coordinates of just one focus and vertex?

Yes. Ellipses are symmetrical, so the coordinates of the vertices of an ellipse centered around the origin will always have the form   ( ± a , 0 )   or   ( 0 ,   ± a ) .   Similarly, the coordinates of the foci will always have the form   ( ± c , 0 )   or   ( 0 ,   ± c ) .   Knowing this, we can use   a   and   c   from the given points, along with the equation   c 2 = a 2 − b 2 , to find   b 2 .

Writing equations of ellipses not centered at the origin

Like the graphs of other equations, the graph of an ellipse    can be translated. If an ellipse is translated   h   units horizontally and   k   units vertically, the center of the ellipse will be   ( h , k ) .   This translation results in the standard form of the equation we saw previously, with   x   replaced by   ( x − h )   and y replaced by   ( y − k ) .

Questions & Answers

differentiate between demand and supply giving examples
Lambiv Reply
differentiated between demand and supply using examples
Lambiv
what is labour ?
Lambiv
how will I do?
Venny Reply
how is the graph works?I don't fully understand
Rezat Reply
information
Eliyee
devaluation
Eliyee
t
WARKISA
hi guys good evening to all
Lambiv
multiple choice question
Aster Reply
appreciation
Eliyee
explain perfect market
Lindiwe Reply
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
Ezea
What is ceteris paribus?
Shukri Reply
other things being equal
AI-Robot
When MP₁ becomes negative, TP start to decline. Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Kelo
yes,thank you
Shukri
Can I ask you other question?
Shukri
what is monopoly mean?
Habtamu Reply
What is different between quantity demand and demand?
Shukri Reply
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
Ezea
ok
Shukri
how do you save a country economic situation when it's falling apart
Lilia Reply
what is the difference between economic growth and development
Fiker Reply
Economic growth as an increase in the production and consumption of goods and services within an economy.but Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
production function means
Jabir
What do you think is more important to focus on when considering inequality ?
Abdisa Reply
any question about economics?
Awais Reply
sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
thank you so much 👍 sir
Asui
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50. A,Calculate quantities of x and y which maximize utility. B,Calculate value of Lagrange multiplier. C,Calculate quantities of X and Y consumed with a given price. D,alculate optimum level of output .
Feyisa Reply
Answer
Feyisa
c
Jabir
the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema
Gsbwnw Reply
suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product
Abdureman
types of unemployment
Yomi Reply
What is the difference between perfect competition and monopolistic competition?
Mohammed
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 7

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College algebra' conversation and receive update notifications?

Ask