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Rewriting r equals 6 over the quantity 2 plus 3 cos theta in standard form gives which eccentricity and curve?
Ae is 1.5, so a hyperbola
Be is 3.0, so a hyperbola
Ce is 0.5, so an ellipse
De is 1.0, so a parabola
Answer & Solution
Correct answer: A. e is 1.5, so a hyperbola
1. Standard form needs the denominator to begin with 1, so divide top and bottom by 2.
2. The numerator 6 divided by 2 becomes 3.
3. The denominator 2 plus 3 cos theta divided by 2 becomes 1 plus 1.5 cos theta.
4. The eccentricity is the coefficient sitting in front of the cosine, so e is 1.5.
5. Since 1.5 is greater than 1, the curve is a hyperbola.
6. Reading e as 3.0 takes the original coefficient without dividing the denominator through.
7. Reading e as 0.5 inverts the ratio and would wrongly promise a closed curve.
8. Reading e as 1.0 would require the cosine coefficient to match the leading 1 exactly.
_Source: OpenStax College Algebra (CC BY 4.0), Ch 8 "Analytic Geometry", section 8.5 Conic Sections in Polar Coordinates_