Question

Eccentricities and DirectricesExercises 29–36 give the eccentricities of conic sections with one focus at the origin along with the directrix corresponding to that focus. Find a polar equation for each conic section.e = 2, x = 4

Accepted Answer

Recall the general polar form of a conic section with a focus at the origin:

$$ r = \frac{ed}{1 + e \cos 	heta} $$

where $$e is $$the eccentricity, $$d is $$the distance from the focus to the directrix, and $$	heta is $$the polar angle. Identify the given values: eccentricity $$e = 2$$ and the directrix is the vertical line $$x = 4. $$Since the directrix is vertical and to the right of the origin, the directrix is located at $$x = d = 4. $$Because the directrix is vertical and to the right, the polar equation uses $$\cos 	heta in $$the denominator with a negative sign if the directrix is to the right (positive $$x-$$axis). The formula becomes:

$$ r = \frac{ed}{1 - e \cos 	heta} $$ Substitute the known values $$e = 2$$ and $$d = 4$$ into the formula:

$$ r = \frac{2 	imes 4}{1 - 2 \cos 	heta} = \frac{8}{1 - 2 \cos 	heta} $$ This expression represents the polar equation of the conic section with eccentricity 2 and directrix $$x = 4. $$Note that since $$e \u003e 1$$, the conic is a hyperbola.

Eccentricities and Directrices

Exercises 29–36 give the eccentricities of conic sections with one focus at the origin along with the directrix corresponding to that focus. Find a polar equation for each conic section.

e = 2, x = 4

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Key Concepts

Eccentricity of Conic Sections

Directrix and Its Role in Polar Equations

Polar Equation of Conic Sections