Identify each conic using eccentricity. (a) r=41+3sin⁡θ (b) r=71−3cos⁡θ (c) r=86+5cos⁡θ (d) r=32−3sin⁡θ

Step 1 Let us find eccentricity of given function We have to find eccentricity if e=1, the conic is a parabola if e=0, the conic is a circle if e&lt;1, the conic us an ellipse if e&gt;1, the conic is a hyperbola r=ed1±esin⁡θ r=ed1±ecos⁡θ Step 2 Solution will be at end of step 2 a)r=41+3sin⁡θ r=41+3sin⁡θ e=3 (e&gt;1, the conic is a hyperbola) b) r=71−cos⁡θ e=1(e=1, the conic is a parabola) Step 3 c) r=86+5cos⁡θ r=86(1+56cos⁡θ) r=86(1+56cos⁡θ) r=43(1+56cos⁡θ) e=56 e=0.83, e&lt;1 (the conic is an ellipse) d) r=32−3sin⁡θ r=32(1−32sin⁡θ) r=322(1−32sin⁡θ) e=32 e=1.5 (e&gt;1, the conic is a hyperbola) Answer: a) e=3 b) e=1 c) e=0.83 d) e=1.5

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Bevan Mcdonald

Answered question

2021-08-14

Identify each conic using eccentricity.
(a) $r = \frac{4}{1 + 3 \sin θ}$
(b) $r = \frac{7}{1 - 3 \cos θ}$
(c) $r = \frac{8}{6 + 5 \cos θ}$
(d) $r = \frac{3}{2 - 3 \sin θ}$

Answer & Explanation

Ezra Herbert

Skilled2021-08-15Added 99 answers

Identifying:

Vasquez

Expert2021-12-27Added 669 answers

Step 1

Let us find eccentricity of given function

We have to find eccentricity

if e=1, the conic is a parabola

if e=0, the conic is a circle

if e<1, the conic us an ellipse

if e>1, the conic is a hyperbola

$r = \frac{e d}{1 \pm e \sin θ}$

$r = \frac{e d}{1 \pm e \cos θ}$

Step 2

Solution will be at end of step 2

a) $r = \frac{4}{1 + 3 \sin θ}$

$r = \frac{4}{1 + 3 \sin θ}$

e=3 (e>1, the conic is a hyperbola)

b) $r = \frac{7}{1 - \cos θ}$

e=1(e=1, the conic is a parabola)

Step 3

c) $r = \frac{8}{6 + 5 \cos θ}$

$r = \frac{8}{6 (1 + \frac{5}{6} \cos θ)}$

$r = \frac{\frac{8}{6}}{(1 + \frac{5}{6} \cos θ)}$

$r = \frac{\frac{4}{3}}{(1 + \frac{5}{6} \cos θ)}$

$e = \frac{5}{6}$

e=0.83, e<1 (the conic is an ellipse)

d) $r = \frac{3}{2 - 3 \sin θ}$

$r = \frac{3}{2 (1 - \frac{3}{2} \sin θ)}$

$r = \frac{\frac{3}{2}}{2 (1 - \frac{3}{2} \sin θ)}$

$e = \frac{3}{2}$

e=1.5 (e>1, the conic is a hyperbola)

Answer:

a) e=3

b) e=1

c) e=0.83

d) e=1.5

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