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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

Cos45/sec30+cosec30=
dinesh Reply
Cos 45 = 1/ √ 2 sec 30 = 2/√3 cosec 30 = 2. =1/√2 / 2/√3+2 =1/√2/2+2√3/√3 =1/√2*√3/2+2√3 =√3/√2(2+2√3) =√3/2√2+2√6 --------- (1) =√3 (2√6-2√2)/((2√6)+2√2))(2√6-2√2) =2√3(√6-√2)/(2√6)²-(2√2)² =2√3(√6-√2)/24-8 =2√3(√6-√2)/16 =√18-√16/8 =3√2-√6/8 ----------(2)
exercise 1.2 solution b....isnt it lacking
Miiro Reply
I dnt get dis work well
john Reply
what is one-to-one function
Iwori Reply
what is the procedure in solving quadratic equetion at least 6?
Qhadz Reply
Almighty formula or by factorization...or by graphical analysis
Damian
I need to learn this trigonometry from A level.. can anyone help here?
wisdom Reply
yes am hia
Miiro
tanh2x =2tanhx/1+tanh^2x
Gautam Reply
cos(a+b)+cos(a-b)/sin(a+b)-sin(a-b)=cotb ... pls some one should help me with this..thanks in anticipation
favour Reply
f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
Ken Reply
proof
AUSTINE
sebd me some questions about anything ill solve for yall
Manifoldee Reply
cos(a+b)+cos(a-b)/sin(a+b)-sin(a-b)= cotb
favour
how to solve x²=2x+8 factorization?
Kristof Reply
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
SO THE ANSWER IS X=-8
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
i am in
Cliff
1KI POWER 1/3 PLEASE SOLUTIONS
Prashant Reply
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
Reuben Reply
which of these functions is not uniformly cintinuous on (0, 1)? sinx
Pooja Reply
helo
Akash
hlo
Akash
Hello
Hudheifa
which of these functions is not uniformly continuous on 0,1
Basant Reply
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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