# 10.5 Conic sections in polar coordinates  (Page 3/8)

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## Graphing a hyperbola in polar form

Graph

First, we rewrite the conic in standard form by multiplying the numerator and denominator by the reciprocal of 2, which is $\text{\hspace{0.17em}}\frac{1}{2}.$

Because $\text{\hspace{0.17em}}e=\frac{3}{2},e>1,$ so we will graph a hyperbola    with a focus at the origin. The function has a term and there is a subtraction sign in the denominator, so the directrix is $\text{\hspace{0.17em}}y=-p.$

The directrix is $\text{\hspace{0.17em}}y=-\frac{8}{3}.$

Plotting a few key points as in [link] will enable us to see the vertices. See [link] .

A B C D
$\theta$ $0$ $\frac{\pi }{2}$ $\pi$ $\frac{3\pi }{2}$
$r=\frac{8}{2-3\mathrm{sin}\text{\hspace{0.17em}}\theta }$ $4$ $-8$ $4$ $\frac{8}{5}=1.6$

## Graphing an ellipse in polar form

Graph

First, we rewrite the conic in standard form by multiplying the numerator and denominator by the reciprocal of 5, which is $\text{\hspace{0.17em}}\frac{1}{5}.$

Because $\text{\hspace{0.17em}}e=\frac{4}{5},e<1,$ so we will graph an ellipse    with a focus at the origin. The function has a $\text{\hspace{0.17em}}\text{cos}\text{\hspace{0.17em}}\theta ,$ and there is a subtraction sign in the denominator, so the directrix    is $\text{\hspace{0.17em}}x=-p.$

The directrix is $\text{\hspace{0.17em}}x=-\frac{5}{2}.$

Plotting a few key points as in [link] will enable us to see the vertices. See [link] .

A B C D
$\theta$ $0$ $\frac{\pi }{2}$ $\pi$ $\frac{3\pi }{2}$
$10$ $2$ $\frac{10}{9}\approx 1.1$ $2$

Graph

## Deﬁning conics in terms of a focus and a directrix

So far we have been using polar equations of conics to describe and graph the curve. Now we will work in reverse; we will use information about the origin, eccentricity, and directrix to determine the polar equation.

Given the focus, eccentricity, and directrix of a conic, determine the polar equation.

1. Determine whether the directrix is horizontal or vertical. If the directrix is given in terms of $\text{\hspace{0.17em}}y,$ we use the general polar form in terms of sine. If the directrix is given in terms of $\text{\hspace{0.17em}}x,$ we use the general polar form in terms of cosine.
2. Determine the sign in the denominator. If $\text{\hspace{0.17em}}p<0,$ use subtraction. If $\text{\hspace{0.17em}}p>0,$ use addition.
3. Write the coefficient of the trigonometric function as the given eccentricity.
4. Write the absolute value of $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ in the numerator, and simplify the equation.

## Finding the polar form of a vertical conic given a focus at the origin and the eccentricity and directrix

Find the polar form of the conic given a focus at the origin, $\text{\hspace{0.17em}}e=3\text{\hspace{0.17em}}$ and directrix     $\text{\hspace{0.17em}}y=-2.$

The directrix is $\text{\hspace{0.17em}}y=-p,$ so we know the trigonometric function in the denominator is sine.

Because $\text{\hspace{0.17em}}y=-2,–2<0,$ so we know there is a subtraction sign in the denominator. We use the standard form of

and $\text{\hspace{0.17em}}e=3\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}|-2|=2=p.$

Therefore,

## Finding the polar form of a horizontal conic given a focus at the origin and the eccentricity and directrix

Find the polar form of a conic given a focus at the origin, $\text{\hspace{0.17em}}e=\frac{3}{5},$ and directrix     $\text{\hspace{0.17em}}x=4.$

Because the directrix is $\text{\hspace{0.17em}}x=p,$ we know the function in the denominator is cosine. Because $\text{\hspace{0.17em}}x=4,4>0,$ so we know there is an addition sign in the denominator. We use the standard form of

and $\text{\hspace{0.17em}}e=\frac{3}{5}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}|4|=4=p.$

Therefore,

$\begin{array}{l}\begin{array}{l}\hfill \\ \hfill \\ r=\frac{\left(\frac{3}{5}\right)\left(4\right)}{1+\frac{3}{5}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta }\hfill \end{array}\hfill \\ r=\frac{\frac{12}{5}}{1+\frac{3}{5}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta }\hfill \\ r=\frac{\frac{12}{5}}{1\left(\frac{5}{5}\right)+\frac{3}{5}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta }\hfill \\ r=\frac{\frac{12}{5}}{\frac{5}{5}+\frac{3}{5}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta }\hfill \\ r=\frac{12}{5}\cdot \frac{5}{5+3\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta }\hfill \\ r=\frac{12}{5+3\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta }\hfill \end{array}$

Find the polar form of the conic given a focus at the origin, $\text{\hspace{0.17em}}e=1,$ and directrix $\text{\hspace{0.17em}}x=-1.$

$r=\frac{1}{1-\mathrm{cos}\theta }$

#### Questions & Answers

The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
divide simplify each answer 3/2÷5/4
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations