# 12.1 The ellipse  (Page 4/16)

 Page 4 / 16

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

The standard form of the equation of an ellipse with center and major axis    parallel to the x -axis is

$\frac{{\left(x-h\right)}^{2}}{{a}^{2}}+\frac{{\left(y-k\right)}^{2}}{{b}^{2}}=1$

where

• $a>b$
• the length of the major axis is $\text{\hspace{0.17em}}2a$
• the coordinates of the vertices are $\text{\hspace{0.17em}}\left(h±a,k\right)$
• the length of the minor axis is $\text{\hspace{0.17em}}2b$
• the coordinates of the co-vertices are $\text{\hspace{0.17em}}\left(h,k±b\right)$
• the coordinates of the foci are $\text{\hspace{0.17em}}\left(h±c,k\right),$ 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 $\text{\hspace{0.17em}}\left(h,k\right)\text{\hspace{0.17em}}$ and major axis parallel to the y -axis is

$\frac{{\left(x-h\right)}^{2}}{{b}^{2}}+\frac{{\left(y-k\right)}^{2}}{{a}^{2}}=1$

where

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

Just as with ellipses centered at the origin, ellipses that are centered at a point $\text{\hspace{0.17em}}\left(h,k\right)\text{\hspace{0.17em}}$ have vertices, co-vertices, and foci that are related by the equation $\text{\hspace{0.17em}}{c}^{2}={a}^{2}-{b}^{2}.\text{\hspace{0.17em}}$ We can use this relationship along with the midpoint and distance formulas to find the equation of the ellipse in standard form when the vertices and foci are given.

Given the vertices and foci of an ellipse not centered at the origin, write its equation in standard form.

1. Determine whether the major axis is parallel to the x - or y -axis.
1. If the y -coordinates of the given vertices and foci are the same, then the major axis is parallel to the x -axis. Use the standard form $\text{\hspace{0.17em}}\frac{{\left(x-h\right)}^{2}}{{a}^{2}}+\frac{{\left(y-k\right)}^{2}}{{b}^{2}}=1.$
2. If the x -coordinates of the given vertices and foci are the same, then the major axis is parallel to the y -axis. Use the standard form $\text{\hspace{0.17em}}\frac{{\left(x-h\right)}^{2}}{{b}^{2}}+\frac{{\left(y-k\right)}^{2}}{{a}^{2}}=1.$
2. Identify the center of the ellipse $\text{\hspace{0.17em}}\left(h,k\right)\text{\hspace{0.17em}}$ using the midpoint formula and the given coordinates for the vertices.
3. Find $\text{\hspace{0.17em}}{a}^{2}\text{\hspace{0.17em}}$ by solving for the length of the major axis, $\text{\hspace{0.17em}}2a,$ which is the distance between the given vertices.
4. Find $\text{\hspace{0.17em}}{c}^{2}\text{\hspace{0.17em}}$ using $\text{\hspace{0.17em}}h\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}k,$ found in Step 2, along with the given coordinates for the foci.
5. Solve for $\text{\hspace{0.17em}}{b}^{2}\text{\hspace{0.17em}}$ using the equation $\text{\hspace{0.17em}}{c}^{2}={a}^{2}-{b}^{2}.$
6. Substitute the values for $\text{\hspace{0.17em}}h,k,{a}^{2},$ 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 a point other than the origin

What is the standard form equation of the ellipse that has vertices $\text{\hspace{0.17em}}\left(-2,-8\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(-2,\text{2}\right)$

and foci $\text{\hspace{0.17em}}\left(-2,-7\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(-2,\text{1}\right)?$

The x -coordinates of the vertices and foci are the same, so the major axis is parallel to the y -axis. Thus, the equation of the ellipse will have the form

$\frac{{\left(x-h\right)}^{2}}{{b}^{2}}+\frac{{\left(y-k\right)}^{2}}{{a}^{2}}=1$

First, we identify the center, $\text{\hspace{0.17em}}\left(h,k\right).\text{\hspace{0.17em}}$ The center is halfway between the vertices, $\text{\hspace{0.17em}}\left(-2,-8\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(-2,\text{2}\right).\text{\hspace{0.17em}}$ Applying the midpoint formula, we have:

Next, we find $\text{\hspace{0.17em}}{a}^{2}.\text{\hspace{0.17em}}$ The length of the major axis, $\text{\hspace{0.17em}}2a,$ is bounded by the vertices. We solve for $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ by finding the distance between the y -coordinates of the vertices.

$\begin{array}{c}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}2a=2-\left(-8\right)\\ 2a=10\\ a=5\end{array}$

So $\text{\hspace{0.17em}}{a}^{2}=25.$

Now we find $\text{\hspace{0.17em}}{c}^{2}.\text{\hspace{0.17em}}$ The foci are given by $\text{\hspace{0.17em}}\left(h,k±c\right).\text{\hspace{0.17em}}$ So, $\text{\hspace{0.17em}}\left(h,k-c\right)=\left(-2,-7\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(h,k+c\right)=\left(-2,\text{1}\right).\text{\hspace{0.17em}}$ We substitute $\text{\hspace{0.17em}}k=-3\text{\hspace{0.17em}}$ using either of these points to solve for $\text{\hspace{0.17em}}c.$

$\begin{array}{c}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}k+c=1\\ -3+c=1\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}c=4\end{array}$

So $\text{\hspace{0.17em}}{c}^{2}=16.$

Next, we solve for $\text{\hspace{0.17em}}{b}^{2}\text{\hspace{0.17em}}$ using the equation $\text{\hspace{0.17em}}{c}^{2}={a}^{2}-{b}^{2}.$

$\begin{array}{c}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{c}^{2}={a}^{2}-{b}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}16=25-{b}^{2}\\ {b}^{2}=9\end{array}$

Finally, we substitute the values found for $\text{\hspace{0.17em}}h,k,{a}^{2},$ and $\text{\hspace{0.17em}}{b}^{2}\text{\hspace{0.17em}}$ into the standard form equation for an ellipse:

$\text{\hspace{0.17em}}\frac{{\left(x+2\right)}^{2}}{9}+\frac{{\left(y+3\right)}^{2}}{25}=1$

what are you up to?
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
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
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
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
Jeffrey
Jeffrey
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?
which equation
Miranda
I dont know. lol
Jeffrey
Miranda
Jeffrey
answer and questions in exercise 11.2 sums
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 .
cos(- z)=cos z
Mustafa
what is a algebra
(x+x)3=?
6x
Obed
what is the identity of 1-cos²5x equal to?
__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
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
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
I don't understand how radicals works pls
How look for the general solution of a trig function