<< 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

write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
B=Ai-itan(hx-hiy)
Rukmini
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
archana Reply
the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
Practice Key Terms 7

Get the best Algebra and trigonometry course in your pocket!





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