Standard forms of the equation of an ellipse with center (0,0)
The standard form of the equation of an ellipse with center
and major axis on the
x-axis is
where
the length of the major axis is
the coordinates of the vertices are
the length of the minor axis is
the coordinates of the co-vertices are
the coordinates of the foci are
, where
See
[link]a
The standard form of the equation of an ellipse with center
and major axis on the
y-axis is
where
the length of the major axis is
the coordinates of the vertices are
the length of the minor axis is
the coordinates of the co-vertices are
the coordinates of the foci are
, where
See
[link]b
Note that the vertices, co-vertices, and foci are related by the equation
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.
Determine whether the major axis lies on the
x - or
y -axis.
If the given coordinates of the vertices and foci have the form
and
respectively, then the major axis is the
x -axis. Use the standard form
If the given coordinates of the vertices and foci have the form
and
respectively, then the major axis is the
y -axis. Use the standard form
Use the equation
along with the given coordinates of the vertices and foci, to solve for
Substitute the values for
and
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
and foci
The foci are on the
x -axis, so the major axis is the
x -axis. Thus, the equation will have the form
The vertices are
so
and
The foci are
so
and
We know that the vertices and foci are related by the equation
Solving for
we have:
Now we need only substitute
and
into the standard form of the equation. The equation of the ellipse is
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
or
Similarly, the coordinates of the foci will always have the form
or
Knowing this, we can use
and
from the given points, along with the equation
to find
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
units horizontally and
units vertically, the center of the ellipse will be
This
translation results in the standard form of the equation we saw previously, with
replaced by
and
y replaced by
Questions & Answers
for the "hiking" mix, there are 1,000 pieces in the mix, containing 390.8 g of fat, and 165 g of protein. if there is the same amount of almonds as cashews, how many of each item is in the trail mix?
an object is traveling around a circle with a radius of 13 meters .if in 20 seconds a central angle of 1/7 Radian is swept out what are the linear and angular speed of the object
like this: (2)/(2-x)
the aim is to see what will not be compatible with this rational expression. If x= 0 then the fraction is undefined since we cannot divide by zero. Therefore, the domain consist of all real numbers except 2.
functions can be understood without a lot of difficulty.
Observe the following:
f(2) 2x - x
2(2)-2= 2
now observe this:
(2,f(2)) ( 2, -2)
2(-x)+2 = -2
-4+2=-2
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
100•3=300
300=50•2^x
6=2^x
x=log_2(6)
=2.5849625
so, 300=50•2^2.5849625
and, so,
the # of bacteria will double every (100•2.5849625) =
258.49625 minutes
Thomas
158.5
This number can be developed by using algebra and logarithms.
Begin by moving log(2) to the right hand side of the equation like this:
t/100 log(2)= log(3)
step 1: divide each side by log(2)
t/100=1.58496250072
step 2: multiply each side by 100 to isolate t.
t=158.49
Dan
what is the importance knowing the graph of circular functions?
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
how to find x:
12x = 144
notice how 12 is being multiplied by x. Therefore division is needed to isolate x
and whatever we do to one side of the equation we must do to the other.
That develops this:
x= 144/12
divide 144 by 12 to get x.
addition:
12+x= 14
subtract 12 by each side. x =2
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.