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
I'm interested in biological psychology and cognitive psychology
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
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miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills