Deriving the equation of an ellipse centered at the origin
Let
and
be the
foci of a hyperbola centered at the origin. The hyperbola is the set of all points
such that the difference of the distances from
to the foci is constant. See
[link] .
If
is a vertex of the hyperbola, the distance from
to
is
The distance from
to
is
The sum of the distances from the foci to the vertex is
If
is a point on the hyperbola, we can define the following variables:
By definition of a hyperbola,
is constant for any point
on the hyperbola. We know that the difference of these distances is
for the vertex
It follows that
for any point on the hyperbola. As with the derivation of the equation of an ellipse, we will begin by applying the
distance formula . The rest of the derivation is algebraic. Compare this derivation with the one from the previous section for ellipses.
This equation defines a hyperbola centered at the origin with vertices
and co-vertices
Standard forms of the equation of a hyperbola with center (0,0)
The standard form of the equation of a hyperbola with center
and transverse axis on the
x -axis is
Note that the vertices, co-vertices, and foci are related by the equation
When we are given the equation of a hyperbola, we can use this relationship to identify its vertices and foci.
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