<< Chapter < Page Chapter >> Page >

What is the standard form equation of the ellipse that has vertices ( −3 , 3 ) and ( 5 , 3 ) and foci ( 1 2 3 , 3 ) and ( 1 + 2 3 , 3 ) ?

( x 1 ) 2 16 + ( y 3 ) 2 4 = 1

Got questions? Get instant answers now!

Graphing ellipses centered at the origin

Just as we can write the equation for an ellipse given its graph, we can graph an ellipse given its equation. To graph ellipses centered at the origin, we use the standard form x 2 a 2 + y 2 b 2 = 1 ,   a > b for horizontal ellipses and x 2 b 2 + y 2 a 2 = 1 ,   a > b for vertical ellipses.

Given the standard form of an equation for an ellipse centered at ( 0 , 0 ) , sketch the graph.

  1. Use the standard forms of the equations of an ellipse to determine the major axis, vertices, co-vertices, and foci.
    1. If the equation is in the form x 2 a 2 + y 2 b 2 = 1 , where a > b , then
      • the major axis is the x -axis
      • the coordinates of the vertices are ( ± a , 0 )
      • the coordinates of the co-vertices are ( 0, ± b )
      • the coordinates of the foci are ( ± c , 0 )
    2. If the equation is in the form x 2 b 2 + y 2 a 2 = 1 , where a > b , then
      • the major axis is the y -axis
      • the coordinates of the vertices are ( 0, ± a )
      • the coordinates of the co-vertices are ( ± b , 0 )
      • the coordinates of the foci are ( 0, ± c )
  2. Solve for c using the equation c 2 = a 2 b 2 .
  3. Plot the center, vertices, co-vertices, and foci in the coordinate plane, and draw a smooth curve to form the ellipse.

Graphing an ellipse centered at the origin

Graph the ellipse given by the equation, x 2 9 + y 2 25 = 1. Identify and label the center, vertices, co-vertices, and foci.

First, we determine the position of the major axis. Because 25 > 9 , the major axis is on the y -axis. Therefore, the equation is in the form x 2 b 2 + y 2 a 2 = 1 , where b 2 = 9 and a 2 = 25. It follows that:

  • the center of the ellipse is ( 0 , 0 )
  • the coordinates of the vertices are ( 0, ± a ) = ( 0, ± 25 ) = ( 0, ± 5 )
  • the coordinates of the co-vertices are ( ± b , 0 ) = ( ± 9 , 0 ) = ( ± 3 , 0 )
  • the coordinates of the foci are ( 0, ± c ) , where c 2 = a 2 b 2 Solving for c , we have:

c = ± a 2 b 2 = ± 25 9 = ± 16 = ± 4

Therefore, the coordinates of the foci are ( 0, ± 4 ) .

Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse. See [link] .

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

Graph the ellipse given by the equation x 2 36 + y 2 4 = 1. Identify and label the center, vertices, co-vertices, and foci.

center: ( 0 , 0 ) ; vertices: ( ± 6 , 0 ) ; co-vertices: ( 0 , ± 2 ) ; foci: ( ± 4 2 , 0 )

Got questions? Get instant answers now!

Graphing an ellipse centered at the origin from an equation not in standard form

Graph the ellipse given by the equation 4 x 2 + 25 y 2 = 100. Rewrite the equation in standard form. Then identify and label the center, vertices, co-vertices, and foci.

First, use algebra to rewrite the equation in standard form.

  4 x 2 + 25 y 2 = 100    4 x 2 100 + 25 y 2 100 = 100 100          x 2 25 + y 2 4 = 1

Next, we determine the position of the major axis. Because 25 > 4 , the major axis is on the x -axis. Therefore, the equation is in the form x 2 a 2 + y 2 b 2 = 1 , where a 2 = 25 and b 2 = 4. It follows that:

  • the center of the ellipse is ( 0 , 0 )
  • the coordinates of the vertices are ( ± a , 0 ) = ( ± 25 , 0 ) = ( ± 5 , 0 )
  • the coordinates of the co-vertices are ( 0, ± b ) = ( 0, ± 4 ) = ( 0, ± 2 )
  • the coordinates of the foci are ( ± c , 0 ) , where c 2 = a 2 b 2 . Solving for c , we have:

c = ± a 2 b 2 = ± 25 4 = ± 21

Therefore the coordinates of the foci are ( ± 21 , 0 ) .

Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse.

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

Questions & Answers

the diagram of the digestive system
Assiatu Reply
How does twins formed
William Reply
They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together
Oluwatobi
what is genetics
Josephine Reply
Genetics is the study of heredity
Misack
how does twins formed?
Misack
What is manual
Hassan Reply
discuss biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles
Joseph Reply
what is biology
Yousuf Reply
the study of living organisms and their interactions with one another and their environments
AI-Robot
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
Joseph Reply
what is the blood cells
Shaker Reply
list any five characteristics of the blood cells
Shaker
lack electricity and its more savely than electronic microscope because its naturally by using of light
Abdullahi Reply
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
Ramadan
What is classification
ISCONT Reply
is organisms that are similar into groups called tara
Yamosa
in what situation (s) would be the use of a scanning electron microscope be ideal and why?
Kenna Reply
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
Hilary
cell is the building block of life.
Condoleezza Reply
what is cell divisoin?
Aron Reply
Diversity of living thing
ISCONT
what is cell division
Aron Reply
Cell division is the process by which a single cell divides into two or more daughter cells. It is a fundamental process in all living organisms and is essential for growth, development, and reproduction. Cell division can occur through either mitosis or meiosis.
AI-Robot
What is life?
Allison Reply
life is defined as any system capable of performing functions such as eating, metabolizing,excreting,breathing,moving,Growing,reproducing,and responding to external stimuli.
Mohamed
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 7

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