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
$\text{\hspace{0.17em}}e\text{\hspace{0.17em}}$ as the coefficient of the trigonometric function in the denominator.
Compare
$\text{\hspace{0.17em}}e\text{\hspace{0.17em}}$ with 1 to determine the shape of the conic.
Determine the directrix as
$\text{\hspace{0.17em}}x=p\text{\hspace{0.17em}}$ if cosine is in the denominator and
$\text{\hspace{0.17em}}y=p\text{\hspace{0.17em}}$ if sine is in the denominator. Set
$\text{\hspace{0.17em}}ep\text{\hspace{0.17em}}$ equal to the numerator in standard form to solve for
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ or
$\text{\hspace{0.17em}}y.$
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,
$\text{\hspace{0.17em}}\frac{1}{c},$ where
$\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ is that constant.
Multiply the numerator and denominator by
$\text{\hspace{0.17em}}\frac{1}{3}.$
Because
$\mathrm{sin}\text{}\theta $ is in the denominator, the directrix is
$\text{\hspace{0.17em}}y=p.\text{\hspace{0.17em}}$ Comparing to standard form, note that
$\text{\hspace{0.17em}}e=\frac{2}{3}.$ Therefore, from the numerator,
Since
$\text{\hspace{0.17em}}e<1,$ the conic is an
ellipse . The eccentricity is
$\text{\hspace{0.17em}}e=\frac{2}{3}$ and the directrix is
$\text{\hspace{0.17em}}y=3.$
Multiply the numerator and denominator by
$\text{\hspace{0.17em}}\frac{1}{4}.$
Because
$\text{cos}\text{\hspace{0.17em}}\theta $ is in the denominator, the directrix is
$\text{\hspace{0.17em}}x=p.\text{\hspace{0.17em}}$ Comparing to standard form,
$\text{\hspace{0.17em}}e=\frac{5}{4}.\text{\hspace{0.17em}}$ Therefore, from the numerator,
Since
$\text{\hspace{0.17em}}e>1,$ the conic is a
hyperbola . The eccentricity is
$\text{\hspace{0.17em}}e=\frac{5}{4}\text{\hspace{0.17em}}$ and the directrix is
$\text{\hspace{0.17em}}x=\frac{12}{5}=\mathrm{2.4.}$
Multiply the numerator and denominator by
$\text{\hspace{0.17em}}\frac{1}{2}.$
Because sine is in the denominator, the directrix is
$\text{\hspace{0.17em}}y=-p.\text{\hspace{0.17em}}$ Comparing to standard form,
$\text{\hspace{0.17em}}e=1.\text{\hspace{0.17em}}$ Therefore, from the numerator,
Because
$\text{\hspace{0.17em}}e=1,$ the conic is a
parabola . The eccentricity is
$\text{\hspace{0.17em}}e=1\text{\hspace{0.17em}}$ and the directrix is
$\text{\hspace{0.17em}}y=-\frac{7}{2}=\mathrm{-3.5.}$
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
$\text{\hspace{0.17em}}e\text{\hspace{0.17em}}$ and, therefore, the shape of the curve. The next step is to substitute values for
$\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ and solve for
$\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ to plot a few key points. Setting
$\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ equal to
$\text{\hspace{0.17em}}0,\frac{\pi}{2},\pi ,$ and
$\text{\hspace{0.17em}}\frac{3\pi}{2}\text{\hspace{0.17em}}$ provides the vertices so we can create a rough sketch of the graph.
First, we rewrite the conic in standard form by multiplying the numerator and denominator by the reciprocal of 3, which is
$\text{\hspace{0.17em}}\frac{1}{3}.$
Because
$\text{\hspace{0.17em}}e=1,$ we will graph a
parabola with a focus at the origin. The function has a
$\mathrm{cos}\text{}\theta ,$ and there is an addition sign in the denominator, so the directrix is
$\text{\hspace{0.17em}}x=p.$
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5) and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.