Given the polar equation for a conic, identify the type of conic, the directrix, and the eccentricity.
Multiply the numerator and denominator by the reciprocal of the constant in the denominator to rewrite the equation in standard form.
Identify the eccentricity
as the coefficient of the trigonometric function in the denominator.
Compare
with 1 to determine the shape of the conic.
Determine the directrix as
if cosine is in the denominator and
if sine is in the denominator. Set
equal to the numerator in standard form to solve for
or
Identifying a conic given the polar form
For each of the following equations, identify the conic with focus at the origin, the
directrix , and the
eccentricity .
For each of the three conics, we will rewrite the equation in standard form. Standard form has a 1 as the constant in the denominator. Therefore, in all three parts, the first step will be to multiply the numerator and denominator by the reciprocal of the constant of the original equation,
where
is that constant.
Multiply the numerator and denominator by
Because
is in the denominator, the directrix is
Comparing to standard form, note that
Therefore, from the numerator,
Since
the conic is an
ellipse . The eccentricity is
and the directrix is
Multiply the numerator and denominator by
Because
is in the denominator, the directrix is
Comparing to standard form,
Therefore, from the numerator,
Since
the conic is a
hyperbola . The eccentricity is
and the directrix is
Multiply the numerator and denominator by
Because sine is in the denominator, the directrix is
Comparing to standard form,
Therefore, from the numerator,
Because
the conic is a
parabola . The eccentricity is
and the directrix is
When graphing in Cartesian coordinates, each conic section has a unique equation. This is not the case when graphing in polar coordinates. We must use the eccentricity of a conic section to determine which type of curve to graph, and then determine its specific characteristics. The first step is to rewrite the conic in standard form as we have done in the previous example. In other words, we need to rewrite the equation so that the denominator begins with 1. This enables us to determine
and, therefore, the shape of the curve. The next step is to substitute values for
and solve for
to plot a few key points. Setting
equal to
and
provides the vertices so we can create a rough sketch of the graph.
Graphing a parabola in polar form
Graph
First, we rewrite the conic in standard form by multiplying the numerator and denominator by the reciprocal of 3, which is
Because
we will graph a
parabola with a focus at the origin. The function has a
and there is an addition sign in the denominator, so the directrix is
The directrix is
Plotting a few key points as in
[link] will enable us to see the vertices. See
[link] .
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