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Graphing a hyperbola in polar form

Graph r = 8 2 3   sin   θ .

First, we rewrite the conic in standard form by multiplying the numerator and denominator by the reciprocal of 2, which is 1 2 .

r = 8 2 3 sin   θ = 8 ( 1 2 ) 2 ( 1 2 ) 3 ( 1 2 ) sin   θ r = 4 1 3 2   sin   θ

Because e = 3 2 , e > 1 , so we will graph a hyperbola    with a focus at the origin. The function has a sin   θ term and there is a subtraction sign in the denominator, so the directrix is y = p .

       4 = e p        4 = ( 3 2 ) p 4 ( 2 3 ) = p       8 3 = p

The directrix is y = 8 3 .

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

A B C D
θ 0 π 2 π 3 π 2
r = 8 2 3 sin θ 4 8 4 8 5 = 1.6
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Graphing an ellipse in polar form

Graph r = 10 5 4   cos   θ .

First, we rewrite the conic in standard form by multiplying the numerator and denominator by the reciprocal of 5, which is 1 5 .

r = 10 5 4 cos   θ = 10 ( 1 5 ) 5 ( 1 5 ) 4 ( 1 5 ) cos   θ r = 2 1 4 5   cos   θ

Because e = 4 5 , e < 1 , so we will graph an ellipse    with a focus at the origin. The function has a cos θ , and there is a subtraction sign in the denominator, so the directrix    is x = p .

       2 = e p        2 = ( 4 5 ) p 2 ( 5 4 ) = p       5 2 = p

The directrix is x = 5 2 .

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

A B C D
θ 0 π 2 π 3 π 2
r = 10 5 4   cos   θ 10 2 10 9 1.1 2
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Graph r = 2 4 cos   θ .

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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 y , we use the general polar form in terms of sine. If the directrix is given in terms of x , we use the general polar form in terms of cosine.
  2. Determine the sign in the denominator. If p < 0 , use subtraction. If p > 0 , use addition.
  3. Write the coefficient of the trigonometric function as the given eccentricity.
  4. Write the absolute value of p 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, e = 3 and directrix     y = 2.

The directrix is y = p , so we know the trigonometric function in the denominator is sine.

Because y = −2 , –2 < 0 , so we know there is a subtraction sign in the denominator. We use the standard form of

r = e p 1 e   sin   θ

and e = 3 and | −2 | = 2 = p .

Therefore,

r = ( 3 ) ( 2 ) 1 3   sin   θ r = 6 1 3   sin   θ
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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, e = 3 5 , and directrix     x = 4.

Because the directrix is x = p , we know the function in the denominator is cosine. Because x = 4 , 4 > 0 , so we know there is an addition sign in the denominator. We use the standard form of

r = e p 1 + e   cos   θ

and e = 3 5 and | 4 | = 4 = p .

Therefore,

r = ( 3 5 ) ( 4 ) 1 + 3 5 cos θ r = 12 5 1 + 3 5 cos θ r = 12 5 1 ( 5 5 ) + 3 5 cos θ r = 12 5 5 5 + 3 5 cos θ r = 12 5 5 5 + 3 cos θ r = 12 5 + 3 cos θ
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Find the polar form of the conic given a focus at the origin, e = 1 , and directrix x = −1.

r = 1 1 cos θ

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Questions & Answers

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Mark Reply
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Miranda
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Miranda Drice
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Miranda
I am living in india
jai
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Miranda
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Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
sita Reply
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
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jai
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Propessor
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Pearl Reply
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
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Jeffrey
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Miranda
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Jeffrey
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Jeffrey
I'm going to 11grade
Miranda
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Miranda
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Steve
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Steve
I don't know why. But Im trying to like it.
Jeffrey
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Jeffrey
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Miranda
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Nelson Reply
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Miranda
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Jeffrey
answer and questions in exercise 11.2 sums
Yp Reply
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
Swadesh
cos(- z)=cos z
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what is a algebra
Jallah Reply
(x+x)3=?
Narad
6x
Obed
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liyemaikhaya Reply
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C'est comment
Abdel
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SORIE
it's 12
what is the function of sine with respect of cosine , graphically
Karl Reply
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Steve
cosx.cos2x.cos4x.cos8x
Aashish Reply
sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
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How look for the general solution of a trig function
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Practice Key Terms 2

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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