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
Discuss the differences between taste and flavor, including how other sensory inputs contribute to our perception of flavor.
The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include:
1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
asegid
to transport fluids fats proteins and lymphocytes to the blood stream as lymph
Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
Kamara
yes
Prince
how does the stomach protect itself from the damaging effects of HCl
the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body
the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature