<< Chapter < Page Chapter >> Page >

Graph the ellipse given by the equation 49 x 2 + 16 y 2 = 784. Rewrite the equation in standard form. Then identify and label the center, vertices, co-vertices, and foci.

Standard form: x 2 16 + y 2 49 = 1 ; center: ( 0 , 0 ) ; vertices: ( 0 , ± 7 ) ; co-vertices: ( ± 4 , 0 ) ; foci: ( 0 , ± 33 )

Got questions? Get instant answers now!

Graphing ellipses not centered at the origin

When an ellipse    is not centered at the origin, we can still use the standard forms to find the key features of the graph. When the ellipse is centered at some point, ( h , k ) , we use the standard forms ( x h ) 2 a 2 + ( y k ) 2 b 2 = 1 ,   a > b for horizontal ellipses and ( x h ) 2 b 2 + ( y k ) 2 a 2 = 1 ,   a > b for vertical ellipses. From these standard equations, we can easily determine the center, vertices, co-vertices, foci, and positions of the major and minor axes.

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

  1. Use the standard forms of the equations of an ellipse to determine the center, position of the major axis, vertices, co-vertices, and foci.
    1. If the equation is in the form ( x h ) 2 a 2 + ( y k ) 2 b 2 = 1 , where a > b , then
      • the center is ( h , k )
      • the major axis is parallel to the x -axis
      • the coordinates of the vertices are ( h ± a , k )
      • the coordinates of the co-vertices are ( h , k ± b )
      • the coordinates of the foci are ( h ± c , k )
    2. If the equation is in the form ( x h ) 2 b 2 + ( y k ) 2 a 2 = 1 , where a > b , then
      • the center is ( h , k )
      • the major axis is parallel to the y -axis
      • the coordinates of the vertices are ( h , k ± a )
      • the coordinates of the co-vertices are ( h ± b , k )
      • the coordinates of the foci are ( h , k ± 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 ( h , k )

Graph the ellipse given by the equation, ( x + 2 ) 2 4 + ( y 5 ) 2 9 = 1. Identify and label the center, vertices, co-vertices, and foci.

First, we determine the position of the major axis. Because 9 > 4 , the major axis is parallel to the y -axis. Therefore, the equation is in the form ( x h ) 2 b 2 + ( y k ) 2 a 2 = 1 , where b 2 = 4 and a 2 = 9. It follows that:

  • the center of the ellipse is ( h , k ) = ( −2 , 5 )
  • the coordinates of the vertices are ( h , k ± a ) = ( 2 , 5 ± 9 ) = ( 2 , 5 ± 3 ) , or ( −2 , 2 ) and ( −2 , 8 )
  • the coordinates of the co-vertices are ( h ± b , k ) = ( 2 ± 4 , 5 ) = ( 2 ± 2 , 5 ) , or ( −4 , 5 ) and ( 0 , 5 )
  • the coordinates of the foci are ( h , k ± c ) , where c 2 = a 2 b 2 . Solving for c , we have:
c = ± a 2 b 2 = ± 9 4 = ± 5

Therefore, the coordinates of the foci are ( −2 , 5 5 ) and ( −2 , 5+ 5 ) .

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!

Graph the ellipse given by the equation ( x 4 ) 2 36 + ( y 2 ) 2 20 = 1. Identify and label the center, vertices, co-vertices, and foci.

Center: ( 4 , 2 ) ; vertices: ( 2 , 2 ) and ( 10 , 2 ) ; co-vertices: ( 4 , 2 2 5 ) and ( 4 , 2 + 2 5 ) ; foci: ( 0 , 2 ) and ( 8 , 2 )

Got questions? Get instant answers now!

Given the general form of an equation for an ellipse centered at ( h , k ), express the equation in standard form.

  1. Recognize that an ellipse described by an equation in the form a x 2 + b y 2 + c x + d y + e = 0 is in general form.
  2. Rearrange the equation by grouping terms that contain the same variable. Move the constant term to the opposite side of the equation.
  3. Factor out the coefficients of the x 2 and y 2 terms in preparation for completing the square.
  4. Complete the square for each variable to rewrite the equation in the form of the sum of multiples of two binomials squared set equal to a constant, m 1 ( x h ) 2 + m 2 ( y k ) 2 = m 3 , where m 1 , m 2 , and m 3 are constants.
  5. Divide both sides of the equation by the constant term to express the equation in standard form.

Questions & Answers

Why is b in the answer
Dahsolar Reply
how do you work it out?
Brad Reply
answer
Ernest
heheheehe
Nitin
(Pcos∅+qsin∅)/(pcos∅-psin∅)
John Reply
how to do that?
Rosemary Reply
what is it about?
Amoah
how to answer the activity
Chabelita Reply
how to solve the activity
Chabelita
solve for X,,4^X-6(2^)-16=0
Alieu Reply
x4xminus 2
Lominate
sobhan Singh jina uniwarcity tignomatry ka long answers tile questions
harish Reply
t he silly nut company makes two mixtures of nuts: mixture a and mixture b. a pound of mixture a contains 12 oz of peanuts, 3 oz of almonds and 1 oz of cashews and sells for $4. a pound of mixture b contains 12 oz of peanuts, 2 oz of almonds and 2 oz of cashews and sells for $5. the company has 1080
ZAHRO Reply
If  , , are the roots of the equation 3 2 0, x px qx r     Find the value of 1  .
Swetha Reply
Parts of a pole were painted red, blue and yellow. 3/5 of the pole was red and 7/8 was painted blue. What part was painted yellow?
Patrick Reply
Parts of the pole was painted red, blue and yellow. 3 /5 of the pole was red and 7 /8 was painted blue. What part was painted yellow?
Patrick
how I can simplify algebraic expressions
Katleho Reply
Lairene and Mae are joking that their combined ages equal Sam’s age. If Lairene is twice Mae’s age and Sam is 69 yrs old, what are Lairene’s and Mae’s ages?
Mary Reply
23yrs
Yeboah
lairenea's age is 23yrs
ACKA
hy
Katleho
Ello everyone
Katleho
Laurene is 46 yrs and Mae is 23 is
Solomon
hey people
christopher
age does not matter
christopher
solve for X, 4^x-6(2*)-16=0
Alieu
prove`x^3-3x-2cosA=0 (-π<A<=π
Mayank Reply
create a lesson plan about this lesson
Rose Reply
Excusme but what are you wrot?
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, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask