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Find the polar equation of an ellipse with its focus at the pole and vertices at (−1,0) and (3,0).

Sagot :

The polar equation of an ellipse is [tex]r=-\frac{3}{1+2cos\theta}[/tex].

The vertices of ellipse are (−1,0) and (3,0).

The polar equation of an ellipse can be represented as

[tex]r=\frac{ep}{1+ecos\theta}[/tex]

where e is the eccentricity.

Eccentricity, e = [tex]\frac{c}{a}[/tex]

c is the distance from the center to the focus and a is the distance from the center to the vertex

[tex]c=\frac{3-(-1)}{2}[/tex]

⇒ [tex]c=\frac{4}{2}[/tex]

⇒ c = 2

[tex]a=\frac{3+(-1)}{2}[/tex]

⇒ [tex]a=\frac{2}{2}[/tex]

⇒ a = 1

Then, e = [tex]\frac{2}{1}[/tex]

⇒ e = 2

Now, the polar equation of an ellipse becomes as,

⇒ [tex]r=\frac{2p}{1+2cos\theta}[/tex] ------- (1)

Now plug in a vertex point such as (-1,0) and solve for p,

⇒ [tex]-1=\frac{2p}{1+2cos0}[/tex]

⇒ [tex]-1=\frac{2p}{1+2(1)}[/tex]               [∵ cos 0 = 1]

⇒ [tex]-1=\frac{2p}{3}[/tex]

⇒ [tex]-3=2p[/tex]

⇒ [tex]p=-\frac{3}{2}[/tex]

Thus the polar equation of an ellipse (1) becomes as,

⇒ [tex]r=\frac{2(-\frac{3}{2} )}{1+2cos\theta}[/tex]

⇒ [tex]r=-\frac{3}{1+2cos\theta}[/tex]

Hence we can conclude that the polar equation of an ellipse is [tex]r=-\frac{3}{1+2cos\theta}[/tex].

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