<< Chapter < Page Chapter >> Page >

Deriving the equation of an ellipse centered at the origin

To derive the equation of an ellipse    centered at the origin, we begin with the foci ( c , 0 ) and ( c , 0 ) . The ellipse is the set of all points ( x , y ) such that the sum of the distances from ( x , y ) to the foci is constant, as shown in [link] .

If ( a , 0 ) is a vertex    of the ellipse, the distance from ( c , 0 ) to ( a , 0 ) is a ( c ) = a + c . The distance from ( c , 0 ) to ( a , 0 ) is a c . The sum of the distances from the foci    to the vertex is

( a + c ) + ( a c ) = 2 a

If ( x , y ) is a point on the ellipse, then we can define the following variables:

d 1 = the distance from  ( c , 0 )  to  ( x , y ) d 2 = the distance from  ( c , 0 )  to  ( x , y )

By the definition of an ellipse, d 1 + d 2 is constant for any point ( x , y ) on the ellipse. We know that the sum of these distances is 2 a for the vertex ( a , 0 ) . It follows that d 1 + d 2 = 2 a for any point on the ellipse. We will begin the derivation by applying the distance formula. The rest of the derivation is algebraic.

                                       d 1 + d 2 = ( x ( c ) ) 2 + ( y 0 ) 2 + ( x c ) 2 + ( y 0 ) 2 = 2 a Distance formula ( x + c ) 2 + y 2 + ( x c ) 2 + y 2 = 2 a Simplify expressions .                              ( x + c ) 2 + y 2 = 2 a ( x c ) 2 + y 2 Move radical to opposite side .                                ( x + c ) 2 + y 2 = [ 2 a ( x c ) 2 + y 2 ] 2 Square both sides .                      x 2 + 2 c x + c 2 + y 2 = 4 a 2 4 a ( x c ) 2 + y 2 + ( x c ) 2 + y 2 Expand the squares .                      x 2 + 2 c x + c 2 + y 2 = 4 a 2 4 a ( x c ) 2 + y 2 + x 2 2 c x + c 2 + y 2 Expand remaining squares .                                               2 c x = 4 a 2 4 a ( x c ) 2 + y 2 2 c x Combine like terms .                                     4 c x 4 a 2 = 4 a ( x c ) 2 + y 2 Isolate the radical .                                         c x a 2 = a ( x c ) 2 + y 2 Divide by 4 .                                     [ c x a 2 ] 2 = a 2 [ ( x c ) 2 + y 2 ] 2 Square both sides .                      c 2 x 2 2 a 2 c x + a 4 = a 2 ( x 2 2 c x + c 2 + y 2 ) Expand the squares .                      c 2 x 2 2 a 2 c x + a 4 = a 2 x 2 2 a 2 c x + a 2 c 2 + a 2 y 2 Distribute  a 2 .                   a 2 x 2 c 2 x 2 + a 2 y 2 = a 4 a 2 c 2 Rewrite .                     x 2 ( a 2 c 2 ) + a 2 y 2 = a 2 ( a 2 c 2 ) Factor common terms .                                x 2 b 2 + a 2 y 2 = a 2 b 2 Set  b 2 = a 2 c 2 .                              x 2 b 2 a 2 b 2 + a 2 y 2 a 2 b 2 = a 2 b 2 a 2 b 2 Divide both sides by  a 2 b 2 .                                       x 2 a 2 + y 2 b 2 = 1 Simplify .

Thus, the standard equation of an ellipse is x 2 a 2 + y 2 b 2 = 1. This equation defines an ellipse centered at the origin. If a > b , the ellipse is stretched further in the horizontal direction, and if b > a , the ellipse is stretched further in the vertical direction.

Writing equations of ellipses centered at the origin in standard form

Standard forms of equations tell us about key features of graphs. Take a moment to recall some of the standard forms of equations we’ve worked with in the past: linear, quadratic, cubic, exponential, logarithmic, and so on. By learning to interpret standard forms of equations, we are bridging the relationship between algebraic and geometric representations of mathematical phenomena.

The key features of the ellipse    are its center, vertices , co-vertices , foci    , and lengths and positions of the major and minor axes . Just as with other equations, we can identify all of these features just by looking at the standard form of the equation. There are four variations of the standard form of the ellipse. These variations are categorized first by the location of the center (the origin or not the origin), and then by the position (horizontal or vertical). Each is presented along with a description of how the parts of the equation relate to the graph. Interpreting these parts allows us to form a mental picture of the ellipse.

Questions & Answers

what are odd numbers
micheal Reply
numbers that leave a remainder when divided by 2
Thorben
1,3,5,7,... 99,...867
Thorben
the third and the seventh terms of a G.P are 81 and 16, find the first and fifth terms.
Suleiman Reply
if a=3, b =4 and c=5 find the six trigonometric value sin
Martin Reply
pls how do I factorize x⁴+x³-7x²-x+6=0
Gift Reply
in a function the input value is called
Rimsha Reply
how do I test for values on the number line
Modesta Reply
if a=4 b=4 then a+b=
Rimsha Reply
a+b+2ab
Kin
commulative principle
DIOSDADO
a+b= 4+4=8
Mimi
If a=4 and b=4 then we add the value of a and b i.e a+b=4+4=8.
Tariq
what are examples of natural number
sani Reply
an equation for the line that goes through the point (-1,12) and has a slope of 2,3
Katheryn Reply
3y=-9x+25
Ishaq
show that the set of natural numberdoes not from agroup with addition or multiplication butit forms aseni group with respect toaaddition as well as multiplication
Komal Reply
x^20+x^15+x^10+x^5/x^2+1
Urmila Reply
evaluate each algebraic expression. 2x+×_2 if ×=5
Sarch Reply
if the ratio of the root of ax+bx+c =0, show that (m+1)^2 ac =b^2m
Awe Reply
By the definition, is such that 0!=1.why?
Unikpel Reply
(1+cosA+IsinA)(1+cosB+isinB)/(cos@+isin@)(cos$+isin$)
Ajay Reply
hatdog
Mark
jaks
Ryan
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