# 12.1 The ellipse  (Page 3/16)

 Page 3 / 16

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

The standard form of the equation of an ellipse with center $\text{\hspace{0.17em}}\left(0,0\right)\text{\hspace{0.17em}}$ and major axis on the x-axis is

$\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$

where

• $a>b$
• the length of the major axis is $\text{\hspace{0.17em}}2a$
• the coordinates of the vertices are $\text{\hspace{0.17em}}\left(±a,0\right)$
• the length of the minor axis is $\text{\hspace{0.17em}}2b$
• the coordinates of the co-vertices are $\text{\hspace{0.17em}}\left(0,±b\right)$
• the coordinates of the foci are $\text{\hspace{0.17em}}\left(±c,0\right)$ , where $\text{\hspace{0.17em}}{c}^{2}={a}^{2}-{b}^{2}.\text{\hspace{0.17em}}$ See [link] a

The standard form of the equation of an ellipse with center $\text{\hspace{0.17em}}\left(0,0\right)\text{\hspace{0.17em}}$ and major axis on the y-axis is

$\frac{{x}^{2}}{{b}^{2}}+\frac{{y}^{2}}{{a}^{2}}=1$

where

• $a>b$
• the length of the major axis is $\text{\hspace{0.17em}}2a$
• the coordinates of the vertices are $\text{\hspace{0.17em}}\left(0,±a\right)$
• the length of the minor axis is $\text{\hspace{0.17em}}2b$
• the coordinates of the co-vertices are $\text{\hspace{0.17em}}\left(±b,0\right)$
• the coordinates of the foci are $\text{\hspace{0.17em}}\left(0,±c\right)$ , where $\text{\hspace{0.17em}}{c}^{2}={a}^{2}-{b}^{2}.\text{\hspace{0.17em}}$ See [link] b

Note that the vertices, co-vertices, and foci are related by the equation $\text{\hspace{0.17em}}{c}^{2}={a}^{2}-{b}^{2}.\text{\hspace{0.17em}}$ 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.

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 $\text{\hspace{0.17em}}\left(±a,0\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(±c,0\right)\text{\hspace{0.17em}}$ respectively, then the major axis is the x -axis. Use the standard form $\text{\hspace{0.17em}}\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1.$
2. If the given coordinates of the vertices and foci have the form $\text{\hspace{0.17em}}\left(0,±a\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(±c,0\right),$ respectively, then the major axis is the y -axis. Use the standard form $\text{\hspace{0.17em}}\frac{{x}^{2}}{{b}^{2}}+\frac{{y}^{2}}{{a}^{2}}=1.$
2. Use the equation $\text{\hspace{0.17em}}{c}^{2}={a}^{2}-{b}^{2},\text{\hspace{0.17em}}$ along with the given coordinates of the vertices and foci, to solve for $\text{\hspace{0.17em}}{b}^{2}.$
3. Substitute the values for $\text{\hspace{0.17em}}{a}^{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{b}^{2}\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}\left(±8,0\right)\text{\hspace{0.17em}}$ and foci $\text{\hspace{0.17em}}\left(±5,0\right)?\text{\hspace{0.17em}}$

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

$\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$

The vertices are $\text{\hspace{0.17em}}\left(±8,0\right),$ so $\text{\hspace{0.17em}}a=8\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{a}^{2}=64.$

The foci are $\text{\hspace{0.17em}}\left(±5,0\right),$ so $\text{\hspace{0.17em}}c=5\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{c}^{2}=25.$

We know that the vertices and foci are related by the equation $\text{\hspace{0.17em}}{c}^{2}={a}^{2}-{b}^{2}.\text{\hspace{0.17em}}$ Solving for $\text{\hspace{0.17em}}{b}^{2},$ we have:

Now we need only substitute $\text{\hspace{0.17em}}{a}^{2}=64\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{b}^{2}=39\text{\hspace{0.17em}}$ into the standard form of the equation. The equation of the ellipse is $\text{\hspace{0.17em}}\frac{{x}^{2}}{64}+\frac{{y}^{2}}{39}=1.$

What is the standard form equation of the ellipse that has vertices $\text{\hspace{0.17em}}\left(0,±4\right)\text{\hspace{0.17em}}$ and foci $\text{\hspace{0.17em}}\left(0,±\sqrt{15}\right)?$

${x}^{2}+\frac{{y}^{2}}{16}=1$

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 $\text{\hspace{0.17em}}\left(±a,0\right)\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\left(0,\text{\hspace{0.17em}}±a\right).\text{\hspace{0.17em}}$ Similarly, the coordinates of the foci will always have the form $\text{\hspace{0.17em}}\left(±c,0\right)\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\left(0,\text{\hspace{0.17em}}±c\right).\text{\hspace{0.17em}}$ Knowing this, we can use $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ from the given points, along with the equation $\text{\hspace{0.17em}}{c}^{2}={a}^{2}-{b}^{2},$ to find $\text{\hspace{0.17em}}{b}^{2}.$

## 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 $\text{\hspace{0.17em}}h\text{\hspace{0.17em}}$ units horizontally and $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ units vertically, the center of the ellipse will be $\text{\hspace{0.17em}}\left(h,k\right).\text{\hspace{0.17em}}$ This translation results in the standard form of the equation we saw previously, with $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ replaced by $\text{\hspace{0.17em}}\left(x-h\right)\text{\hspace{0.17em}}$ and y replaced by $\text{\hspace{0.17em}}\left(y-k\right).$

#### Questions & Answers

what are you up to?
Mark Reply
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
sita Reply
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
Pearl Reply
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
haha. already finished college
Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
Nelson Reply
which equation
Miranda
I dont know. lol
Jeffrey
please where is the equation
Miranda
ask nelson. lol
Jeffrey
answer and questions in exercise 11.2 sums
Yp Reply
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
Swadesh
cos(- z)=cos z
Mustafa
what is a algebra
Jallah Reply
(x+x)3=?
Narad
6x
Obed
what is the identity of 1-cos²5x equal to?
liyemaikhaya Reply
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
Hiiii
Chinni
hello
Ranjay
hi
ANSHU
hiiii
Chinni
h r u friends
Chinni
yes
Hassan
so is their any Genius in mathematics here let chat guys and get to know each other's
SORIE
I speak French
Abdel
okay no problem since we gather here and get to know each other
SORIE
hi im stupid at math and just wanna join here
Yaona
lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
SORIE
it's 12
what is the function of sine with respect of cosine , graphically
Karl Reply
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
Aashish Reply
sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply

### Read also:

#### 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?

 By By By Dewey Compton By Candice Butts By Anh Dao By OpenStax By OpenStax By Brooke Delaney By OpenStax By OpenStax By Stephen Voron By George Turner