"Graph the lines and conic sections r=1(1+cos⁡θ)" solved and explained

Kyran Hudson

Answered question

2021-08-12

Graph the lines and conic sections

r = \frac{1}{(1 + \cos θ)}

Latisha Oneil

Skilled2021-08-13Added 100 answers

Step 1
To identify the conic section, write the given equation in the form of

r = \frac{e p}{1 + e \cos (θ)}

where e is the eccentricity.

r = \frac{1}{1 + \cos (θ)}

= \frac{1 (1)}{1 + (1) \cos (θ)}

Therefore,

e = 1

and

p = 1 .

Hence, the given conic is a parabola.
Step 2
To graph the conic substitute

\sqrt{x^{2} + y^{2}}

for r and x for

r \cos θ

in the given equation

r = \frac{1}{1 + \cos (θ)},

and find the cartesian equation of the parabola. The lines present in the graph when

θ = \pm π, \pm 3 π, \pm 5 π, \cdot

r = \frac{1}{1 + \cos θ}

r (1 + \cos θ) = 1

r + r \cos θ = 1

\sqrt{x^{2} + y^{2}} + x = 1

\sqrt{x^{2} + y^{2}} = 1 - x

x^{2} + y^{2} = {(1 - x)}^{2} x^{2} + y - {(1 - x)}^{2} = 0

x^{2} + y^{2} - 1 + 2 x - x^{2} = 0

y^{2} = 1 - 2 x

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