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Converting a conic in polar form to rectangular form

Convert the conic r = 1 5 5 sin θ to rectangular form.

We will rearrange the formula to use the identities   r = x 2 + y 2 , x = r cos θ , and  y = r sin θ .

                           r = 1 5 5 sin θ   r ( 5 5 sin θ ) = 1 5 5 sin θ ( 5 5 sin θ ) Eliminate the fraction .         5 r 5 r sin θ = 1 Distribute .                          5 r = 1 + 5 r sin θ Isolate  5 r .                      25 r 2 = ( 1 + 5 r sin θ ) 2 Square both sides .           25 ( x 2 + y 2 ) = ( 1 + 5 y ) 2 Substitute  r = x 2 + y 2  and  y = r sin θ .         25 x 2 + 25 y 2 = 1 + 10 y + 25 y 2 Distribute and use FOIL .           25 x 2 10 y = 1 Rearrange terms and set equal to 1 .
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Convert the conic r = 2 1 + 2   cos   θ to rectangular form.

4 8 x + 3 x 2 y 2 = 0

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Access these online resources for additional instruction and practice with conics in polar coordinates.

Visit this website for additional practice questions from Learningpod.

Key concepts

  • Any conic may be determined by a single focus, the corresponding eccentricity, and the directrix. We can also define a conic in terms of a fixed point, the focus P ( r , θ ) at the pole, and a line, the directrix, which is perpendicular to the polar axis.
  • A conic is the set of all points e = P F P D , where eccentricity e is a positive real number. Each conic may be written in terms of its polar equation. See [link] .
  • The polar equations of conics can be graphed. See [link] , [link] , and [link] .
  • Conics can be defined in terms of a focus, a directrix, and eccentricity. See [link] and [link] .
  • We can use the identities r = x 2 + y 2 , x = r   cos   θ , and y = r   sin   θ to convert the equation for a conic from polar to rectangular form. See [link] .

Section exercises

Verbal

Explain how eccentricity determines which conic section is given.

If eccentricity is less than 1, it is an ellipse. If eccentricity is equal to 1, it is a parabola. If eccentricity is greater than 1, it is a hyperbola.

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If a conic section is written as a polar equation, what must be true of the denominator?

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If a conic section is written as a polar equation, and the denominator involves sin   θ , what conclusion can be drawn about the directrix?

The directrix will be parallel to the polar axis.

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If the directrix of a conic section is perpendicular to the polar axis, what do we know about the equation of the graph?

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What do we know about the focus/foci of a conic section if it is written as a polar equation?

One of the foci will be located at the origin.

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Algebraic

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity.

r = 6 1 2   cos   θ

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r = 3 4 4   sin   θ

Parabola with e = 1 and directrix 3 4 units below the pole.

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r = 8 4 3   cos   θ

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r = 5 1 + 2   sin   θ

Hyperbola with e = 2 and directrix 5 2 units above the pole.

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r = 16 4 + 3   cos   θ

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r = 3 10 + 10   cos   θ

Parabola with e = 1 and directrix 3 10 units to the right of the pole.

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r = 4 7 + 2   cos   θ

Ellipse with e = 2 7 and directrix 2 units to the right of the pole.

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r ( 1 cos   θ ) = 3

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r ( 3 + 5 sin   θ ) = 11

Hyperbola with e = 5 3 and directrix 11 5 units above the pole.

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r ( 4 5 sin   θ ) = 1

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r ( 7 + 8 cos   θ ) = 7

Hyperbola with e = 8 7 and directrix 7 8 units to the right of the pole.

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Questions & Answers

An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Adu
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
Practice Key Terms 2

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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